The Montenegrin A ademy of S ien es and Arts

University of Montenegro

University of Evora, Portugal

Dorodni yn Computing Centre of RAS

IV International Conferen e on Optimization

Methods and Appli ations

OPTIMIZATION

AND

APPLICATIONS

(OPTIMA-2013)

Petrova , Montenegro, September 2013

ABSTRACTS

Mos ow 2013

UDC 519.658

Pro eedings in lude extended abstra ts of reports presented at the

IV International Conferen e on Optimization Methods and Appli ations

"Optimization and appli ations" (OPTIMA-2013) held in Petrova , Montenegro,

September 2228, 2013.

Edited by V.U. Malkova.

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Organizing Committee

Miloica Jacimovic, Chair, Montenegrin Academy of Sciences and Arts,MontenegroYury G. Evtushenko, Chair, Dorodnicyn Computing Centre of RAS,RussiaVladimir Bushenkov, Chair, University of Evora, Portugal

Izedin Krnic, University of MontenegroOleg Obradovic, University of MontenegroVladimir Jacimovic, University of MontenegroNevena Mijajlovic, University of MontenegroVera Kovacevic-Vujcic, University of BelgradeVesna Dragovic, Montenegrin Academy of Sciences and ArtsAlexander P. Afanasiev, Institute for Information Transmission Problemsof RAS, RussiaAnatoly S. Antipin, Dorodnicyn Computing Centre of RAS, RussiaNatalia K. Burova, Dorodnicyn Computing Centre of RAS, RussiaVladimir A. Garanzha, Dorodnicyn Computing Centre of RAS, RussiaAlexander I. Golikov, Dorodnicyn Computing Centre of RAS, RussiaAlexander Yu. Gornov, Inst. System Dynamics and Control Theory, SBof RAS, RussiaAlexander V. Lotov, Dorodnicyn Computing Centre of RAS, RussiaYuri Nesterov, CORE Universite Catholique de Louvain, BelgiumGueorgui Smirnov, University of Minho, PortugalTatiana Tchemisova, University of Aveiro, PortugalAlexey A. Tretyakov, Dorodnicyn Computing Centre of RAS, RussiaIvetta A. Zonn, Dorodnicyn Computing Centre of RAS, RussiaVitaly G. Zhadan, Dorodnicyn Computing Centre of RAS, RussiaVladimir I. Zubov, Dorodnicyn Computing Centre RAS, Russia

Contents

Vagif Abdullaev Numerical Solution to a Coefficient Inverse Problem

for a system of linear ODEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

K.K. Abgaryan Multiscale computational model of multy-layer semicon-

ductor nanostructures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

Alexander P. Abramov Two-level planning in a model of economy . . . 15

Oleg V. Abramov Algorithms for Reliability Estimation and Optimiza-

tion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

Alexander Afanasyev, Elena Putilina About some of the misconcep-

tions that occur in the numerical solution of optimal control problems . . 18

Kamil Aida-zade, Anar Rahimov Solution to a class of direct and in-

verse problems with non-local initial and boundary conditions for evolution

equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

Anton Anikin, Alexander Gornov, Alexander Andrianov Compu-

tational technologies for Morse potential optimization . . . . . . . . . . . . . . . . . . 22

Maxim Anop, Yaroslava Katueva Parametric reliability optimization

parallel algorithms for discrete nominal set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

Maxim Anop, Evgenii Murashkin Calculation of irreversible deforma-

tion areas at thermal influence and hydrostatic pressure . . . . . . . . . . . . . . . . 26

Anatoly Antipin A boundary-value optimal control problem with termi-

nal conditions in the form of extreme mappings . . . . . . . . . . . . . . . . . . . . . . . . 27

Yegana Ashrafova Calculation and optimization of regimes of fluid flow

in oil pipeline networks of complicated structure . . . . . . . . . . . . . . . . . . . . . . . . 29

Sergey Astrakov Regular covering and packing with circles . . . . . . . . . . . 31

A.Z. Atamuratov, I.E. Mikhailov, L.A. Muravey On the damping

of oscillations of rectangular membranes and plates . . . . . . . . . . . . . . . . . . . . . 32

Maxim Balashov Metric projection in the real Hilbert space . . . . . . . . . . 34

A.S. Buldaev One approach to optimization of nonlinear controlled sys-

tems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

Vladimir Bushenkov, Margarida Ferreira, Ana Ribeiro, Georgi

Smirnov Optimal control of hydroelectric power stations . . . . . . . . . . . . . . 38

4

Sergey Chukanov Destruction of simple asymptotic trajectories in eco-

nomic dynamical optimization models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

Dmitry Denisov, Olga Nekrasova Optimization problem of the defini-

tion of dividend payment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

V.V. Dikusar, P.P. Gusyatnikova Fast and slow dynamic feedbacks . 42

V.V. Dikusar, N.N. Olenev, M. Wojtowicz Optimal Control Prob-

lems with Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

Anna Dorjieva Algorithms for creation of adaptive grids for non-convex

unconstrained optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

Olga Druzhinina, Natalia Petrova Optimal stabilization of multiply

connected controlled systems with using of homogeneous vector functions 47

Pavel Dvurechenskiy, Yurii Nesterov Primal-dual optimization meth-

ods for finding saddle-points in differential games . . . . . . . . . . . . . . . . . . . . . . . 49

Vladimir Elkin Subsystems of Affine Control Systems . . . . . . . . . . . . . . . . 51

Ivan Eremin, Edward Gimadi, Alexander Kelmanov, Artem Py-

atkin, Mikhail Khachay A 2-approximation algorithm for finding the

clique with minimum weight of vertices and edges . . . . . . . . . . . . . . . . . . . . . . 52

Adil Erzin Wireless sensor networks and covering a plane with sectors . 54

Georgy Evdonov Parallel computations in processes of formalization

principles of the method of branches and borders and solving combina-

torial tasks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

Yury Evtushenko, Mikhail Posypkin Towards Deterministic Con-

strained Multiobjective Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

Rashid Faizullin Functions associated with 3-SAT problem. . . . . . . . . . . . 58

Zuhura Gaeva, Alexander Shananin An application of Chebyshev-

Markov-Krein algorithm to the numerical solution of the control problem,

described by the Smoluchowski kinetic equation . . . . . . . . . . . . . . . . . . . . . . . . 59

Alexander Galashov, Alexander Kel’manov A 2-approximation al-

gorithm for a NP-hard problem of searching a family of disjoint vector

subsets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

Shamil Galiev, Maria Lisafina Optimization of packing equal ellipses

into a rectangular . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

5

Alexander Gasnikov, Yurii Nesterov Entropy optimization problem

and new efficient dual method based on the regularization of dual problem 64

Alexander Golikov Piecewise quadratic functions in the methods of lin-

ear optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

Evgeny Golshteyn, Ustav Malkov, Nikolay Sokolov A Numerical

Method for Solving Bimatrix Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

Maxim Golubev Local strong convexity in the Hilbert space . . . . . . . . . . 68

Evgeniy Goncharov An approach to solve the resource-constrained

project scheduling problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

Alexander Gornov, Anna Dorjieva Numerical techniques for optimiza-

tion problems with “expensive” functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

Alexander Gornov, Evgeniya Finkelstein Practical Approximation

Algorithms for Reachable Set of Dynamic Controlled System . . . . . . . . . . . 74

Valentin Gorokhovik General separation theorems of convex sets and

convex optimization problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

Valentin Gorokhovik, Marina A. Trafimovich Classes of positively

homogeneous functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

Yana Grebenyuk, Aleksandr Abramov Delimiting the Central Busi-

ness District – an economic analysis using graph theory . . . . . . . . . . . . . . . . 78

Milojica Jacimovic, Nevena Mijajlovic Ill-posedness and regularized

algorithms for solving quasi-variational inequalities . . . . . . . . . . . . . . . . . . . . . 79

Vladimir Jacimovic Impulsive control of bifurcations under constraints 80

Igor Kaporin, Olga Milyukova Massively parallel algorithms for two-

side scaling of large sparse nonnegative matrices with applications . . . . . . 81

Anatoly Karpenko, Mikhail Posypkin, Anton Rubtsov, Maxim

Sakharov Multi-Memetic Global Optimization Based on the Mind Evo-

lutionary Computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

Alexander Kel’manov, Sergey Khamidullin A 2-approximation poly-

nomial algorithm for a sequence partitioning problem . . . . . . . . . . . . . . . . . . . 85

Alexander Kel’manov, Vladimir Khandeev A randomized algorithm

for a clustering problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

6

Alexander Kel’manov, Liudmila Mikhailova On a sequences summa-

tion problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

Alexander Kel’manov, Semyon Romanchenko A fully polynomial

time approximation scheme for a problem of choosing a vector subset . . . 88

Alexander Kel’manov, Artem Pyatkin On complexity of one maxi-

mum cut problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

Ruben.V. Khachaturov Algorithms for constructing the Lattices of

Cubes and their applicability for solving optimization problems . . . . . . . . . 90

Michael Khachay Boosting approximation algorithm for minimum affine

separating committee problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

Elena Khoroshilova Optimal control of boundary value problems on

both ends of the time interval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

Yuri Kiselev, Sergey Avvakumov Generalized Chaplygin problem . . 96

Alexander Kolokolov, Lubov AfanasyevaOn solving some production

groups formation problems based on discrete optimization . . . . . . . . . . . . . . 98

Ekaterina Konovalova, Anatoly Panyukov The control of equilibrium

at highly monopolized markets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

Vladimir N. Koterov Selection of the optimal operation mode of the

spillway of Boguchanskaya HPP in winter 2012-2013 . . . . . . . . . . . . . . . . . . . . 101

Vladimir Krivonozhko, Finn Førsund, Andrey Lychev Identifying

suspicious efficient units in DEA models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

Vladimir Krivtsov, Victor Soloviev Computing modeling and inves-

tigation of a surface barrier discharge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

Alexander Lazarev, Elena Musatova, Nail Husnullin The problem

of scheduling a freight train as a problem of integer programming. . . . . . 105

Daria Lemtyuzhnikova, Aleksander Sviridenko, Oleg Shcherbina

On improvement of local algorithms for discrete optimization problems . 106

Alexander Lotov Pareto Frontier Visualization for Nonlinear Models and

its Use in Multiobjective Search for Efficient Therapy of HIV Virus Infec-

tion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

Maria Makarova, Nicholas Olenev On firm investment policy model

accounting for employees learning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

7

Vyacheslav Maksimov The method of control models in problems of

tracking of solutions of differential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

V.U. Malkova Parallel Method of Nonuniform Coverings . . . . . . . . . . . . . . 112

Olga Masina On optimal control of motion of transport systems . . . . . . 114

Arsalan Mizhidon, Klara Mizhidon An optimal control problem with

state and mixed constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

Evgeny Molchanov, Alexander Shananin Inverse problems of output

modeling in the conditions of globalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

Yury Morozov Quasi-time-optimal controls for wheeled robot . . . . . . . . . 120

Evgenii Murashkin Reverse and optimisation problems in creep process 122

L.A.Muravey, V.M.Petrov, A.M.Romanenkov Modeling and opti-

mization of ion-beam etching process II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

Dmitry Nazarov Inscribing a cube of maximum volume in a region of

acceptability for optimal parameters sizing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

Angelia Nedic Distributed Optimization in Networked Systems . . . . . . . 127

Yurii Nesterov Universal gradient methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

Lazar Obradovic Regret in Optimal Stopping and It’s Relation to Am-

biguity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

Valeriy Parkhomenko Global climate stabilization controlled by strato-

spheric aerosols injections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

Alexander Pesterev The best ellipsoidal approximation of the attraction

domain in the path following problem for a wheeled robot with constrained

control resource . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

Lev F. Petrov Control of oscillations in essentially nonlinear dynamic

systems with multiple degrees of freedom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

Alexander PlyasunovOn price competition in Stackelberg location-price

model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

Leonid Popov Least Square Method and Improper MP Problems . . . . . 138

I.G. Pospelov On the decomposition of the Ramsey model . . . . . . . . . . . . 140

Artem Pyatkin Edge multicoloring of weighted unicyclic graphs . . . . . . 141

8

Ruslan Sadykov, Alexander Lazarev, Alexey Karpychev Freight

Car Routing Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

Vladimir Savelyev Controllability and optimality of one-dimensional

movement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

Simon Serovajsky Object and morphism symmetry of optimality condi-

tions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

Ilyas Shakenov Comparing of approximation of functional gradient and

gradient of functional approximation for heat conduction optimization

problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

Kanat ShakenovThe recurrent estimating of unknown parameter of PDF

using control parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

Valery ShevchenkoTriangulations of polyhedra and convex cones and re-

alization of their f -vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

Vladimir Skarin Regularization and Correction Methods for Improper

Convex Programming Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

Elena Slavutskaya, Leonid Slavutskii Optimization of the Psychodi-

agnostic Data Processing Using Neural Networks . . . . . . . . . . . . . . . . . . . . . . . 156

Vladimir Srochko Iterative procedures for solving minimax problems . 158

Bo Tian, Mikhail Posypkin BNBTest@HOME - A distributed imple-

mentation of branch and bound method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

Nikolai Tikhomirov, Tatiana Tikhomirova, Damir Ilyasov Opti-

mization strategies for radiation safety at low doses of irradiation including

the cost of diagnosis of oncologic diseases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

Alexey A. Tretyakov P -regularity theory and singular calculus of vari-

ations problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

Aida Valeeva, Yulia Goncharova,, Ivan KoshcheevThree-Dimensional

Extended Vehicle Routing Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

Alexander Vasin, Polina Kartunova On Optimal Design for Capacity

and Electricity Market . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

Gennady Zabudsky, Anton Koval Algorithms for maxmin location

problem on a plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

9

L.A. Zaozerskaya, V.A. Plankova The development of specialized com-

puter knowledge control system using discrete optimization models . . . . . 168

E.S. Zasukhina, V.V. Dikusar Determining parameters in the model

of water transfer in soil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

Vitaly Zhadan Two-stage affine scaling methods for linear semidefinite

programming problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

Vladimir Zubov, Alla Albu On the New Statement of the Optimal

Control Problem of the Metal Crystallization Process . . . . . . . . . . . . . . . . . . . 172

Anna Zykina, Dmitry Zaporozhets, Dmitry Kujanov, Vladimir

Zykin Efficiency of parallel computing in extragradient methods . . . . . . . 173

Alexandre Dolgui, Anton Eremeev, Mikhail Kovalyov, Vyach-

eslav Sigaev On Complexity of Buffers Allocation Problems for Produc-

tion Lines with Unreliable Machines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

Alexey M. Nazarenko, Alexis I. Pospelov, Fedor V. Gubarev

Multi-objective generalization of probability of improvement method . . . . 176

Nikita Perestoronin, Fedor Gubarev, Alexis Pospelov Multicriteria

optimization technique based on local geometry of Pareto set . . . . . . . . . . . 177

Author index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

10

Numerical Solution to a Coefficient Inverse Problemfor a system of linear ODEs

Vagif Abdullaev1

1 Cybernetics Institute of ANAS, Baku, Azerbaijan;

vaqif ab@rambler.ru

We consider a problem of identifying the parameters of a dynamicsystem

x(t) = A(t)x(t) +B(t)C + F (t), t ∈ (t0, T ]. (1)

Here x(t) ∈ Rn is the phase state of the system; C ∈ Rl is the requiredparameter vector; A(t), B(t), F (t) are given matrices of the dimensions(n× n), (n× l), (n× 1), respectively.

It is assumed that to identify the parameter vector C, we have Ncarried out experiments under different initial conditions:

xj(tj0) = xj0 , j = 1, N, (2)

During the experiments, observations over the state of the dynamicsystem are carried out. The following forms of the results of observationsare possible:a) Separated multipoint conditions:

xjν(tji ) = xjνi , ν = 1,mj , i = 1, sj, , j = 1, N. (3)

Here ν = 1,mj are the indices of the observable phase coordinates x(t)

ate the points of time tj1, ..., tjsj under jth experiment for t ∈ [tj0, T

j]; sj

is the number of the observation time in the jth experiment; L =N∑

j=1

mjsj

is the total number of additional observations for the determination of thevector C.b) Non-separated multipoint nonlocal conditions

k∑

i=1

αjix(t

ji ) = βj , j = 1, N. (4)

11

Here the given matrices αji , β

j , j = 1, N , have the dimensions (L×n)and (L× 1), respectively.

For the cases a) and b) it is assumed that L ≥ l.We propose an approach in which the solution to the problem (1) and

(2) is reduced to the solution to independent of each other purpose-builtsupplementary Cauchy problems with respect to a (l+1) system of the nth

order. Using the values of the obtained solutions for the Cauchy problemsat the points of time when the observations are conducted, we obtain asystem of L linear algebraic equations with respect to the vector:

A1C = B1. (5)

Here A1 is the matrix of the dimension L × l, and B1 is the L -dimensional vector. If L = l, then the solution is determined directlyfrom (5): C = A−1

1 B1. If L > l, then the solution to (5) is the vector

C =(

AT1 A1

)

−1B1.

Substituting the obtained numerical values of the vector parameterC into the right-hand side of (1), we already solve an ordinary Cauchyproblem with respect to (1) with initial conditions (2).

The approach proposed in the work, by using the method of lines, isextended onto the case of solution to coefficient inverse problems withrespect to partial differential equations.

In the report, we will present the results of the numerical experimentswhich have been obtained when solving some test problems with obser-vations of the form (3) and (4) for L = l and L > l. We will presentthe results of the numerical experiments and analysis of the consideredinverse problem with errors in the observations.

References

1. K.R. Aida-zade. “ A Numerical method of restroring the parameters of adynamic system”, Kibern. Sistemn. Anal., 40, No. 3, 392–399 (2004).

2. V.M. Abdullaev, K.R. Aida-zade. “On numerical solution to systems of loadedordinary differential equations”, Computational Mathematics and Mathemat-ical Physics, 44, No. 9, 1585–1595 (2004).

12

Multiscale computational model of multy-layersemiconductor nanostructures

K.K. Abgaryan1

1 Dorodnicyn Computing Centre of RAS, Moscow, Russia; kristal83@mail.ru

This paper represents the results of theoretical research with the intentto apply optimization methods in order to build a physical and mathe-matical model of a multi-layer semiconductor nanostructure (MSNS).

As a modified software model of the MSNS was developed, the exist-ing multi-layer nanostructure calculation system was upgraded based oncomparative analysis of output data from theoretical and experimentalresearch by Rzhanov Institute of Semiconductor Physics Siberian Branchof RAS. A multi-scale calculation system was created for MSNS that canemulate semiconductor nanostructures, consider their defects, and cal-culate their conductive properties, as well as other properties in future.Mathematical research was done to examine stability and scope of appli-cation of the resulting model and equations. The project used softwarethat implements created algorithms on a supercomputer at the Joint Su-percomputer Center of the Russian Academy of Sciences (JSCC RAS). Tocreate a modified system of MSNS calculation, comprised of Unit 1 (sta-tistical calculation of metric parameters and structural properties of theMSNS), Unit 2 (dynamic computation of epitaxial structure growth pro-cesses, including nitride transformation, etc.), and Unit 3 (data bank), theproject used a hierarchical sequence for computation algorithm building.

For example, Unit 1 is a set of software modules used for theoreticalcomputation of equilibrium properties of crystal structures, and of MSNSbased on them. The Unit encompasses computation modules built withvarious solid body models, using methods and approximations such as:

— close-packed arrangement model [1–3];— theory of electronic density functionality, implemented with VASP

software pack [4–7];— method of pseudopotentials;— Inverse software pack (computes vehicular properties based on dig-

ital structure), etc.In the multiple scale approach, data computed with lower level models

are used as inputs for upper-level models. For example, the close-packed

13

model can be used to find metric parameters of crystal arrays that arepart of a multilayer semiconductor structure. Feeding the output into theVASP software pack, we can receive accurate data, reproduce the elec-tronic structure (density) of individual crystal arrays and the multilayerstructure as a whole, calculate coherent interfaces for MSNS, piezoelectricand spontaneous polarization in conductive layers, etc. Subsequently, wecan process the outputs from the previous scale level (VASP) and thecalculated electronic structure, in the original Inverse software, using theself-consistent solution of the Schrodinger–Poisson equation, in order toexamine the vehicular properties inside the MSNS conductivity channel.

The close-packed model and the optimization methods used to findmetric parameters of specific crystal arrays are described in detail in [1–3].This paper is going to elaborate the theory that underlies the optimizationmethods for priority calculations based on the density functionality theory.

The author is supported by the Russian Foundation for Basic Research

(project no. 10-08-01263-a).

References

1. V.R. Khachaturov, K.K. Abgaryan, A.V. Bakaev, R.V. Galiulin, V.I. Ka-banovich. “Search through All Possible Crystal Structures with a GivenChemical Formula and Symmetries by the Example of Perovskite”, PreprintNo. 2, Institution of Russian Academy of Sciences Dorodnicyn ComputingCentre of RAS, Moscow, 1997, pp. 6–22.

2. K.K. Abgaryan, V.R. Khachaturov. “Computer Simulation of Stable Struc-tures in Crystal Substances”, Comp. Maths. Math. Phys., 2009, Vol. 45,No. 8, pp. 1517–1530.

3. K.K.Abgaryan ”Application of optimization methods for modeling of closepackings crystal structures”. International conference ”Optimization and ap-plications” (Optima-2009), Petrovac. Montenegro, 2009, pp. 3-4.

4. P. Hohenberg, W. Kohn Phys. Rev. B, 136 (1964), 864.

5. W. Kohn, L. J. Sham Phys. Rev. A, 140 (1965), A1133.

6. G. Kresse, J. Furthmuller. Phys. Rev. B, 54 (1996), 11169.

7. X.Wang, A.Yoshikawa. Progress in Crystal Growth and Characterization ofMaterials, 2004, 48/49, pp. 42–103.

14

Two-level planning in a model of economy

Alexander P. Abramov1

1 Computing Center RAS, Moscow, Russia; apabramov@list.ru

Consider the Leontiev simple dynamic model with two-level system ofplanning. Let A be a technological matrix, x(t) the output vector at stept, xs(t) the vector of total sale, xd(t) the vector of total demand, xp(t)the vector of perspective production plans, and xp(t) the vector of currentproduction plans. Assume that the vector of perspective production plansat step t + 1 is determined by the total sale of products produced atstep t − 2: xpi (t + 1) = kxsi (t − 2), where coefficient k is the same at allsteps. The vector of perspective plans generates the vector xd(t) of totaldemand for products made on at step t: xd(t) = Axp(t+ 1). In turn, thedemand vector generates the vector of current production plans at step t:xp(t) = xd(t). It is supposed that the produced products are distributedin proportion to the quantities of demand. The current plan is correctedin accordance with the most scarce resource. The corrected current plandetermines the vector xs(t− 1) = Axp(t).

Theorem. If matrix A is primitive, vectors x(0) and xp(2) are

strictly positive and inequalities k ≥ (1/λA)3 hold, then this scheme of

two-level planning either brings the system asymptotically to the turnpike,

or keeps the turnpike regime at appropriate values x(0) and xp(2).This means that the sequence of normalized vectors x(t)/‖x(t)‖ as

t → ∞ converges to limit xA/‖xA‖, i.e., to normalized Fobenius vectorof A. The degree of convergence is estimated by the formula

∥

∥

∥(A/λA)

2t−1xp(2)− xA

∥

∥

∥

∞

< Cr2t−1,

where C is some positive constant, the scalar r is bounded by 1 > r >|λn−1|/λA, λA and λn−1 are the Frobenius eigenvalue of the matrix Aand the second eigenvalue of this matrix correspondingly. Comparisonof this estimation with a similar estimation of the convergence speed inthe model where it is used only current planning, shows that use of thetwo-level planning significantly increases the convergence speed.

The author was supported by the Russian Foundation for Basic Research

(project no. 13-07-00730).

15

Algorithms for Reliability Estimation andOptimization

Oleg V. Abramov1

1 Institute for Automation and Control Processes, Far Eastern Branch RAS,

Vladivostok, Russia; abramov@iacp.dvo.ru

This paper proposes the approach and some algorithms for seeking anumerical solution of the parametric optimization problem with a stochas-tic (reliability) criterion. For solving this problem we use a so-called oper-ational/parametric approach based on the computer-aided simulation ofsystem capability and availability, parameters deviations and techniquesof optimal parametric synthesis with respect to reliability criteria [1]. Spe-cial attention is paid to algorithms that reduce the labour content ofparameters optimization problems. Strategies of using the parallel anddistributed processing techniques for purpose of stochastic criterion esti-mation and optimization have been discussed.

In general the reliability optimization problem can be stated as follows[2].

Given the characteristics of random processes X(t) of system param-eters variations, a region of acceptability – Dx and a service time T ,find such a deterministic vector of parameter ratings (nominals) x =(x1r , . . . , xnr) that the reliability

Pr(xr, T ) = Pr

[X1(t)− x1r , . . . , Xn(t)− xnr] ∈ Dx,

∀t ∈ [0, T ]

= max

The practical algorithm of the reliability estimation is based on theconventional Monte Carlo method. The Monte Carlo method approxi-mates Pr(xr , T ) by the ratio of number of acceptable realizations (fallingin region Dx) – Na to the total number of trials – N . Unfortunately,often the region Dx is unknown. It is given only implicitly through sys-tem’s equations and the systems response functions. If we do not knowthe region Dx, then a Monte Carlo evaluation of probability Pr(xr, T ) atparticular nominal value xr requires N system analyses for each trial setof parameter xr. Optimization requires the evaluation of our probabilityPr(xr, T ) for many different values of the nominal values of the parame-ters xr. Therefore to make practical the use of Monte Carlo techniques in

16

reliability-oriented design, it is necessary to reduce the number of systemanalysis required during optimization.

As a solution, the following two-steps technique and the correspondingalgorithms can be used for practical reliability optimization. The firststep consists in replacing the original stochastic criterion with a certaindeterministic one, allowing nearby optimum solutions to be obtained. Itmeans that such a nominal point should be found that would have thelargest distance from the acceptability region margins. At the same timethe next design step must be made if the reliability index that was achievedby using deterministic methods is not high enough. This step is a directprobability optimization, i.e.; algorithms of stochastic optimization shouldbe used here.

Several algorithms for region of acceptability location and approxi-mation [3], reliability estimation (parallel Monte Carlo algorithm) anddiscrete global optimization using parallel and distributed processing arediscussed.

For seeking a numerical solution of the reliability optimization problema network computer-aided program system is proposed.

This work was funded by the Russian Foundation for Basic Research(project no.11-08-98503 r vostok a).

References

1. O.V. Abramov. Reliability-directed parametric synthesis of stochastic sys-tems. Nauka, (1992).

2. O.V. Abramov. ”Parallel algorithms for computing and optimizing reliabilitywith respect to gradual failures”, Automation and Remote Control, vol. 71,No. 7, 1394-1402, (2010).

3. O.V. Abramov, D.A. Nazarov ”Regions of acceptability approximation inreliability design”, Reliability: Theory and Aplications, vol. 7, No. 3, 43-49,(2012).

17

About some of the misconceptions that occur in thenumerical solution of optimal control problems

Alexander Afanasyev1, Elena Putilina2

1 Institute for Systems Analysis RAS, Moscow, Russia; apa@isa.ru2 Institute for Systems Analysis RAS, Moscow, Russia; elena@isa.ru

We discuss the typical errors that occur during the direct applicationof finite difference schemes for solving optimal control problems. Exam-ples are given to illustrate the situation arise. Also there are discussedapproaches to avoid these errors.

Optimal control problem

J [u] =

∫ T

0

F0(x(t), u(t))dt → min, x(t) = F (x(t), u(t)),

x(0), x(T )) ∈ X – boundary conditions, u(t) ∈ U(x(t)) – set of con-trols and its approximation consist on x(t) → x(ti), u(t) → u(ti), i ∈∈ 0, 1, . . . , N are discussed.

The faults connected with incomprehension the concept of needle vari-ations are illustrated by means simplest Liapunov problem

J [u] =

∫ 3

−3

u(t)2(u(t)− 1)(u(t)− 2)dt→ min, x(t) = u(t),

x(0) = 0, t ∈ [−3, 3]

where f(u) = u2(u− 1)(u− 2),

fmin =

0 if u = 0−5 if u ≈ 13

8

fmax = 1, 6 if u ≈5

8

The faults and effects connected with chattering-mode appearance arediscussed with help the problems

J [u] =

∫ T

0

x2(t)dt→ min, x(n)(t) = u(t), |u(t)| ≤ 1,n−1∑

i=0

(x(i)(0))2 6= 0.

and

18

J [u] = T → min, x(n)(t) = u(t), ‖u(t)‖ ≤ 1, ‖x(t)‖ ≥ 1,

n−1∑

i=0

(x(i)(0))2 = 0.

n−1∑

i=0

(x(i)(T ))2 = 0,

where x(·) : [0, T ] → R2, u(·) : [0, T ] → R2.

The above examples are illustrated also inability to use the Hilbertspace approach for optimal control studies.

19

Solution to a class of direct and inverse problemswith non-local initial and boundary conditions for

evolution equations

Kamil Aida-zade1, Anar Rahimov2

1 Azerbaijan State Oil Academy, Baku, Azerbaijan;

kamil aydazade@rambler.ru2 Cybernetics Institute of ANAS, Baku, Azerbaijan; anar r@yahoo.com

In the report, we consider the following inverse problem with respectto a parabolic equation:

∂v (x, t)

∂t= a (x, t)

∂2v (x, t)

∂x2+ a1 (x, t)

∂v (x, t)

∂x+

+a2 (x, t) v (x, t) + f (x, t) + F (x, t) , (1)

(x, t) ∈ Ω = (x, t) : 0 < x < l, 0 < t ≤ T ,

with initial, final, and boundary conditions:

v (x, 0) = ϕ0 (x) , v (x, T ) = ϕT (x) , x ∈ [0, l] , (2)

v (0, t) = ψ0 (t) , v (l, t) = ψ1 (t) , t ∈ [0, T ] , (3)

where F (x, t) = B (x, t)C (x). a (x, t) > 0, a1 (x, t) , a2 (x, t), f (x, t),B (x, t), ϕ0 (x) , ϕT (x), ψ0 (t) , ψ1 (t) are given functions, continuous withrespect to x and t, B (x, t) is differentiable with respect to t, and a2 (x, t) ≤

0, B (x, t) ≥ 0, ∂B(x,t)∂t ≥ 0. The functions ϕ0 (x) , ϕT (x), ψ0 (t) , ψ1 (t)

satisfy the following consistency conditions:

ϕ0 (0) = ψ0 (0) , ϕ0 (l) = ψ1 (0) , ϕT (0) = ψ0 (T ) , ϕT (l) = ψ1 (T ) .

The problem (1)-(3) consists in determining the unknown continu-ous function C (x) and a corresponding solution to the boundary-valueproblemv (x, t), which is twice continuously differentiable with respect tox and continuously differentiable with respect to t for(x, t) ∈ Ω, and sat-isfy the conditions (1)-(3). It is known that the inverse problem (1)-(3)has a unique solution under the above mentioned assumptions.

20

Let us note that the following boundary value problem with nonlocal(integral) initial conditions can be reduced to a particular case of theproblem (1)-(3):

∂u (x, t)

∂t= a (x)

∂2u (x, t)

∂x2+a1 (x)

∂u (x, t)

∂x+a2 (x) u (x, t)+ f (x, t) , (4)

(x, t) ∈ Ω,

k1u (x, 0) +

T∫

0

ekτu (x, τ ) dτ = ϕ0 (x) , x ∈ [0, l] , (5)

u (0, t) = ψ0 (t) , u (l, t) = ψ1 (t) , t ∈ [0, T ] , (6)

where k, k1are given constants and f (x, t) , ϕ0 (x) , ψ0 (t) , ψ1 (t) are givenfunctions.

We consider a coefficient-inverse problem in which the function F (x, t)in the equation (1) is as follows

F (x, t) = C (x, t)B (t) , (7)

where the function C (x, t) is given, and the coefficient B (t) is identified,and instead of the final condition in (2) we use, for example, the followingcondition:

∂v (0, t)

∂x= ψ2 (t) , t ∈ [0, T ] .

A numerical method of solution to the problem (1)-(3), which does notuse any iterative procedures, is proposed in the report. It is based on themethod of lines and reduces the initial problem to a system of ordinarydifferential equations with unknown parameters, to determine which wepropose to use a special representation of the solution based on the work[1].

All the necessary computational schemes, formulae, and results of thecarried out numerical experiments will be given in the report.

References

1. K.R. Ayda-zade. “Numerical method of identification of dynamic system pa-rameters,” J. Inv. Ill-Posed Problems, 13, 201–211 (2005).

21

Computational technologies for Morse potentialoptimization

Anton Anikin1, Alexander Gornov1, Alexander Andrianov2

1 Institute for System Dynamics and Control Theory of SB RAS, Irkutsk,

Russia; anton.anikin@gmail.com, gornov@icc.ru2 Keldysh Institute of Applied Mathematics of RAS, Moscow, Russia;

and@a5.kiam.ru

Unconstrained minimization problem continues to be one of the mainproblems in the theory of optimization. Perfomance of modern computerprovides opportunities for numerical study a number of large-scale prac-tical problems. One of such problem is finding low-energy atomic andmolecular clusters. This article focuses on optimizing of Morse potential:

V (R) = ǫ∑

i<j

eρ0(1−rij/r0)(eρ0(1−rij/r0) − 2), (1)

where R represents a collective coordinate of all internuclear distances(rij), and r0 is the equilibrium distance for which the pair potentialreaches the minimum value ǫ (i.e., the well depth); here, we apply theMorse potential in reduced coordinates by imposing r0 = 1 and ǫ = 1. Inturn, the parameter ρ establishes the range of the potential.

The paper cosider the Morse potential optimizing problem for “largeclusters”, which currently contain more than 200 atoms [1]. The com-plexity of the Morse potential optimization problem is associated withan astronomical increase in the number of local extrema. For example,for the structure with 147 atoms, experts assessments is about 1060 [2].However, modern algorithms, runned on high performance computers cansuccessfully find probable (“putative”) solutions, which, seemingly, is aglobal extremes.

Autors presents a computational technologies for low-energy solutionsfinding, based on the Big-Bang [3] and Basin-Hoping (MSBH) methods.Quasi-Newton L-BFGS method and different versions of conjugate gradi-ent method (Hestenes-Stiefel, Fletcher-Reeves, Polak-Polyak-Ribiere, Liu-Storey, Dai-Yuan, Dai-Liao, Hu-Storey and other [4]) are used for localoptimization. Cauchy’s method used for correction of finded minimum.

22

Implementations of described methods are made with C++ languageand works on Linux, Windows and Mac OS X platforms. Program codewas build and tested with GCC, ICC and CLang compilers. For ran-dom number generation used a number of uniform generators: urand,libc rand and a pseudo-random generators from GSL library [5], whichinclude “Mersenne Twister” (MT19937), ranlux, cmrg, taus and manyother generators.

Implementations of all described methods was tested by big number ofcomputational experiments. During this experiments was founded globalminimums of Morse potential for clusters with size up to 250 atoms andρ = 3. At present time the maximum size of cluster with known mini-mum is 240 atoms, this result can be founded in works of Y. Feng [1] andJ.M.C. Marques [3]. The minimum from J.M.C. Marques now is betterand was obtained by using Big-Bang method on supercomputer. The au-thors obtained this “best-of-known” minimum with MSBH method, usingBig-Bang method, unfortunatly, not allowed us to find global extremumfor this cluster. Also with MSBH method was obtained new “best-of-known” global minimum for cluster with 250 atoms. All computationalexperiments was executed without using of supercomputer.

The authors were supported by the Russian Foundation for Basic Research(project no. 12-07-31118, project no. 13-01-00470).

References

1. Y. Feng, L. Cheng, H. Liu “Putative Global Minimum Structures of MorseClusters as a Function of the Range of the Potential: 161 <= N <= 240,”Phys. Chem. A, 113, No. 49, 13651–13655 (2009).

2. The Cambridge Cluster Database (CCD)http://www-wales.ch.cam.ac.uk/CCD.html

3. J.M.C. Marques, A.A.C.C. Pais and P.E. Abreu “On the use of big-bangmethod to generate low-energy structures of atomic clusters modeled withpair potentials of different ranges,” Comput. Chem., 33, No. 4, 442–452(2013).

4. Neculai Andrei “40 conjugate gradient algorithms for unconstrained optimiza-tion. A survey on their definition,” ICI Technical Report, No. 13 (2008).

5. GNU Scientific Library (GSL)http://www.gnu.org/software/gsl

23

Parametric reliability optimization parallelalgorithms for discrete nominal set

Maxim Anop1, Yaroslava Katueva2

1 Institute of Automation and Control Processes of Russian Academy ofSciences, Vladivostok, Russia; manop@iacp.dvo.ru

2 Institute of Automation and Control Processes of Russian Academy ofSciences, Vladivostok, Russia; gloria@iacp.dvo.ru

In general, technical devices and systems optimal parametric synthesis prob-lem taking into account reliability requirements and parameters degradation isto find nominals that provide the maximum probability of acceptability for apredetermined operating time. It is assumed that the structure (topology) ofthe designed system and its mathematical model are known.

When designing a complex analog electronic equipment functions describingthe projected system are specified using a simulation model of the “black box”type by applying electrical circuits simulation software. This condition does notallow to obtain the optimal solution using the classical differential optimizationmethods. In order to solve the relibility optimization problem it is necessary toapply zero-ordered optimization algorithms.

Nominal set is a discrete nominal set. The solution search in this case isan exhaustive search in nominal internal parameters set. In each point of thatobjective function is calculated. Hill-climbing parallel algorithms and discard-ing subsets that predetermindly do not contain optimal solutions may decreasesolution search time.

Suppose that the solution search process can be performed using k processes.Nominal parameters set is partitioned into nonoverlapping subsets providedthat each j-th process is assigned its own subset of the original set. Thus, eachj-th process calculates the objective function for all elements of relevant subsetand performs the optimal nominal parameters values vector. The results aretransmitted to main process that perform optimal nominal vector of an overallnominal parameters set.

Objective function calculating features involving Monte-Carlo and criticalsection methods [1] and the lack of necessity of its calculation for nominal valuesthat are outside the acceptability region may result in to a serious imbalancein the parallel computing process. Therefore it is proposed to search optimal

24

solution only within the acceptability region.

For more efficient parallel computing capacities using it is proposed to par-tition the nominal parameters set for a large number of small units so that eachof the computing process as be coming to the end of the work with one unitasks the main process of the following unit from the list of available.

Comparative characteristics of the proposed parallel algorithms are given inthe paper.

The authors were supported by the FEB RAS Foundation for Basic Research(project no. 13-III-B-03-25).

References

1. O.V.Abramov and K.S.Katuyev “Effective methods for parametric synthesisof stochastic systems,” First Asian Control Conference, 3, 587-589 (1994).

25

Calculation of irreversible deformation areas atthermal influence and hydrostatic pressure

Maxim Anop1, Evgenii Murashkin1

1 Institute for Automation and Control Processes Far Eastern Branch RussianAcademy of Sciences, Vladivostok, Russia; murashkin@dvo.ru

The level of residual stresses and strains can have a significant affect onexploitation parameters of materials. In case of high temperature gradientsthe influence of generating residual stresses and decreas in the yield stress ofthe material, could be the determining factor to launch a process of irreversibledeformation. To calculate the condition of an elastic-plastic body, which has ac-cumulated irreversible deformations and to evaluate the level of residual stressesin conditions of cooling the body to initial temperature in particular, one needsto estimate a displacement field. The problem of how to evaluate a displacementfield was firstly considered by D.D. Ivlev within the framework of the theoryof an ideal elastic-plastic body. He showed the way to compute displacementin statically determinate problems of a perfect plasticity theory and indicatedthe conditions when it is feasible. The research is devoted to the study ofhow the characteristics of stress-deformed state of material are developed underhigh temperature gradients and influence of residual deformations on the fur-ther process of temperature stress formation. We detected the emerging zonesof plastic yielding in the vicinity of the cylinder surfaces. We calculated theminimum level of the loading pressure and temperature at which plastic flowbegins simultaneously on both boundary surfaces.

This research was supported by the grant of the President of the RussianFederation (MK-776.2012.1) and the grant of the Russian Foundation for BasicResearch (mol a ved 12-01-33064).

26

A boundary-value optimal control problem withterminal conditions in the form of extreme mappings

Anatoly Antipin

Computing Center of Russian Academy of Sciences, Moscow, Russia;asantip@yandex.ru

The problem of computing a fixed point of extreme mapping is the mostcommon mathematical structure which includes almost all of the known problemformulations. Namely, these are problems of convex programming, saddle andgame programming problems, equilibrium programming problems, variationalinequalities, and dynamic game problems.

In general, the problem of computing a fixed point of extreme mapping maybe formulated as

v∗ ∈ ArgminΦ(v∗, w) + ϕ(w), w ∈ W ⊂ Rn, (1)

where W = w ∈ Rn | Aw ≤ a is convex, bounded polyhedron, the function

Φ(v∗, w) + ϕ(w) is convex in w ∈ W , for any fixed value of the variable v ∈ W .The function Φ(v, w) is antisymmetric with respect to its variables, i.e. satisfiesthe condition Φ(v, w) = −Φ(w, v).

Any positive semi-definite function, i.e. subject to the condition

Φ(w,w)− Φ(w, v)−Φ(v, w) + Φ(v, v) ≥ 0 ∀v, w ∈ W ×W,

can be reduced to the form presented in (1). In the regular case the extreme

mapping (1) always has a fixed point.Problems of type (1) often play the role of mathematical models to describe

the various objects in the economy, technology, politics, and scientific research.However, over time the objects are changed under the influence of various factorsin continuously changing environment. Therefore it is interesting to considerthese models in a dynamic form.

We consider the boundary value optimal control problem, where the bound-ary conditions are the fixed points of extreme mappings (1):

v∗0 ∈ ArgminΦ0(v∗0 , w0) + ϕ0(w0), w0 ∈W ⊂ Rn, (2)

v∗1 ∈ ArgminΦ1(v∗1 , w1) + ϕ1(w1), w1 ∈W ⊂ Rn, (3)

27

d

dtv(t) = D(t)v(t) +B(t)u(t), v(t0) = v∗0 , v(t1) = v∗1 , u(t) ∈ U. (4)

Here, the fixed points of extreme inclusions (2), (3) are the initial andterminal conditions of the dynamical system (4). With the choice of controlu(·) ∈ U ⊂ Rs, the trajectory x(t), as a solution to the differential equation (4),takes an object from state (2) to the state (3).

In the paper the saddle method is formulated, and under certain conditionsits convergence to the solution of the problem is proved. In this case, theconvergence in controls is weak, the convergence in direct and dual trajectoriesis strong, and the convergence in terminal variables also is strong.

The author was supported by the Russian Foundation for Basic Research(project no. 12-01-00783) and by the Program ”Leading Scientific Schools”(NSh-5264.2012.1).

References

1. Anatoly Antipin. “Gradient Approach of Computing Fixed Point of Equilib-rium Problems”, Journal of Global Optimization, 24, 285–309 (2002).

2. Anatoly Antipin. “Extra-proximal methods for solving two-person nonzero-sum games”, Mathematical Programming, Ser.B 120, 147–177 (2009).

3. Anatoly Antipin. “Two-person game with Nash equilibrium in optimal controlproblems”, Optimization Letters, 6, No.7, 1349–1378 (2012).

28

Calculation and optimization of regimes of fluid flowin oil pipeline networks of complicated structure

Yegana Ashrafova1

1 Cybernetics Institute of ANAS, Baku, Azerbaijan;y aspirant@yahoo.com

The calculation and optimization problem of regimes of transient processesin oil pipelines of complicated structure is considered in the work, which isdiffer from many cases considered before for separate linear districts of magistralpipelines.

For illustration we consider the pipeline consisting of five sections consistingof two lines with a link between them. The linearized system of differentialequations for the unsteady-state motion of dripping liquid can be written in thefollowing form:

−∂Pk(x, t)

∂x=∂Qk(x, t)

∂t+ 2akQ

k(x, t),

−∂Pk(x, t)

∂t= c2

∂Qk(x, t)

∂x, x ∈ (0, lk), t ∈ [0, T ], k = 1, ..., 5

where P k(x, t), Qk(x, t)– are the pressure and flow rate at the k-th section, c–isthe speed of sound in the medium, ρ is a constant density in the k-th linearsection of length lk of an oil pipeline of diameter dk, γ– kinematic coefficient ofviscosity is independent of pressure and that the condition 2a = 32γ

d2= const is

quite accurate for a laminar flow.There are pumping stations on four nodes of the pipeline, which provide the

required transfer mode:

Q1(0, t) = u1(t), Q3(0, t) = u3(t)

Q4(0, t) = u4(t), Q5(0, t) = u5(t)

The following conditions are fulfilled on the nodes of the network:

P 1(l1, t) = P 3(l3, t) = P 2(l2, t),

Q1(l1, t) +Q2(l2, t) +Q3(l3, t) = 0,

P 4(0, t) = P 5(0, t) = P 2(0, t),

Q4(0, t) +Q5(0, t) +Q2(0, t) = 0.

We consider, the initial state of the process is unknown at t = 0, but functions

Qk(x, 0) = Qk0 , Pk(x, 0) = P k0 (x),

29

k ∈ K = 1, 3, 4, 5 , x ∈ [0, lk],

determining the initial state of the process are belong to some admissible setof functions: D = Qk0 , P k0 (x) : x ∈ [0, lk], k ∈ K, for each of which allthe conditions of existence and uniqueness of solutions of the correspondingboundary value problem are fulfilled.

The problem consists of finding the minimal time of transient process T andthe values of boundary conditions u(t) = (u1(t), u3(t), u4(t), u5(t)), t ∈ (0, T ],in which the functional defined such as:

J(u, T ) = T +1

mesD

∫

D

∑

k∈K

l∫

0

[[Qk(x, T ;u)−QkT ]

2+

+[P k(x, T ;u)− P kT (x)]2]dxdρ(Qk0)dρ(P

k0 ) → min

takes its minimal value. The functional determines a value for the average devia-tion the state of the process at t = T from the given desired state (QkT (x), P

kT (x)),

k ∈ K = 1, 3, 4, 5 for all admissible initial conditions (Qk0(x), Pk0 (x)) ∈ D,

k ∈ K; ρ(Qk0 , ρ(Pk0 ) – are the distributed functions of the initial values on the

set D. The time interval [t0, T ] plays an important role in the investigation ofthe problems as boundary-value, as well as optimal control, in which the stateof the process already does not depend on the initial conditions at t = 0.

We propose to use the iterative methods of optimization of the first order,based on the using of analytical formulas for the gradients of the target func-tional regarding to control functions for the numerical solution of the consideredproblems of optimal control.

The results of numerical experiments for solving the problems of optimalcontrol and their analysis will be presented at the report.

30

Regular covering and packing with circles

Sergey Astrakov1

1 Design Technological Institute of Digital Techniques, Siberian Branch of theRussian Academy of Sciences, Novosibirsk, Russia; astrakov90@gmail.com

Currently, the cover models have applications in the optimization of wirelesssensor networks. It is known [1,2] that the covering of the plane with circles ofthe same radius has a minimum density of D if the centers of circles located atthe vertices of a regular triangular lattice and all regular triangles (tiles) F arecovered equally. Note that in this case we use n = 1 species circles and tile iscovered by three 1/k = 1/6 parts of the circle. Consequently, the model can bedefined in such way: Cov(F | 1 : 3/6) = Cov(F | 1 : 1/2) = Cov(F | 1 : 12).Tile F may be divided into six identical minimum fragment f (rectangulartriangles), each of which is covered with 1/12 part of a circle. Since the minimumfragment is uniquely determined, the covering has got a designation Cov(f | 1 :112). Similarly, the close packing with circles of the same radius Pac(f | 1 : 112)is defined. For n ≥ 2 we can build generic classes of covering and packing:Cov(f | n : 1/k1, 1/k2, . . . , 1/kn) , Pac(f | n : 1/k1, 1/k2, . . . , 1/kn).

The proposed classification allows us to describe the structure of any of theregular model, and the value of n may be considered as its level of difficulty.In each class of covering solved the problem of finding model with a minimumdensity of D, and in the class of packing - model with maximum density d. Inboth cases, the density index is a measure of efficiency. The paper studies theclasses of n = 2, 3, 4. In particular, we show that in the class Cov(f | 2 : 112, 12)there is a model with a record density of D = 3π

5√

3≈ 1.0883, while, in Pac(f |

2 : 112, 12) there is a model with a density d = 0.96243.The author is supported by the Russian Foundation for Basic Research

(project no. 13-07-00139).

References

1. L. Fejes Tuth. Lagerungen in der ebene auf der kugel und im raum, Springer-Verlag, 1953.

2. R. Kershner. “The Number of Circles Covering a Set,” American Journal ofMathematics, 1, v. 1, No. 3, 665–671 (1939).

31

On the damping of oscillations of rectangularmembranes and plates

A.Z. Atamuratov1, I.E. Mikhailov2, L.A. Muravey3

1 “MATI” — Russian State University of Aviation Technology of K.E.Tsiolkovsky, Moscow; gooffydog@mail.ru, pm@mati.ru

2 Dorodnicyn Computing Centre of RAS, Moscow; mikh igor@mail.ru

The report addresses the problem of damping of oscillations of rectangularmembranes and plates with size l1 × l2 with fixed damper small size, placed onthe mechanical object described above in the small area

ω = (x, y) : x ∈ (x0 − ε1, x0 + ε1), y ∈ (y0 − ε2, y0 + ε2).

Earlier strict controllability of string vibrations has been considered in theworks of J. Lagnesse (1983) and D. Russel (1978). This approaches shows thatthe problem of damping string oscillations has a solution but it is difficult touse in practical applications. In this regard other approaches were examinedincluding the use of a moving point and a fixed dampers with small finite size(A. Atamuratov, S. Aslanov, I. Mikhailov, and L. Muravey 2006–2013).

Oscillations of rectangular membrane are described by hyperbolic equation

utt = a2(uxx + uyy) + g(t, x, y),

t > 0, 0 < x < l1, 0 < y < l2, a = const.(1)

The initial conditions are considered known

u(0, x, y) = H0(x, y), ut(0, x, y) = H1(x, y) (2)

On the boundaries impose fixing conditions

u(t, 0, y) = u(t, l1, y) = u(t, x, 0) = u(t, x, l2) = 0 (3)

While small transverse vibrations of an elastic rectangular isotropic platewith constant thickness are described by equation of Jarmen–Lagrange

utt = −a2∆∆u+ g(t, x, y) (4)

The initial conditions the same (2), while boundary conditions

u|Γ = 0, ∆u|Γ = 0. (5)

For both (1) and (4) g(t, x, y) = w(t)1, (x, y) ∈ ω; 0, (x, y) /∈ ω, where w(t)— control function.

32

The problem of damping: to find the control function g(t, x, y) and theminimum time T such that

u(T, x, y) = 0, ut(T, x, y) = 0 (6)

Solving equations (1) and (4) by Fourier method and substituting it inconditions (6), we obtain the following system of moments

T∫0

gn,m(τ ) sin[aλn,m(T − τ)] dτ = L1(λn,m; An,m; Bn,m),

T∫0

gn,m(τ ) cos[aλn,m(T − τ )] dτ = L2(λn,m; An,m; Bn,m),

T∫0

gn,m(τ ) exp(iaλ2n,mτ ) dτ = L1(λ

2n,m; An,m; Bn,m),

T∫0

gn,m(τ ) exp(−iaλ2n,mτ ) dτ = L2(λ

2n,m; An,m; Bn,m),

(8)

where L1, L2, L1, L2 — linear combinations of eigenvalues and Fourier’s coef-ficients for initial conditions (2).

It is shown that for (7) for the system sin(aλn,mt), cos(aλn,mt), λ2n,m =

(πn/l1)2 + (πm/l2)

2, we can use condition of N. Levinson (1940) of Riesz basisin L2(0, T ), therefore this system has biorthgonal system on the interval (0, T ),that allows us to damp oscillations for T =

√πl1l2/a. While for (8) basing on

the works of R. Bellman (1944) and A. Butkovskiy (1975) it can be shown thatthe system exp(iaλ2

n,mt, exp(−iaλ2n,mt), λ2

n,m =√

(πn/l1)4 + (πm/l2)4, isalmost orthogonal, and since the series of expressions in the right side of system(8) are convergent it can be concluded that control function gn,m(t) exists.

In this work for solving and analyzing solutions of the problem of damp-ing we developed numerical algorithms which shows high effectiveness of usingcomputing methods before analytical.

33

Metric projection in the real Hilbert space

Maxim Balashov1

1 Moscow Institute of Physics and Technology, Dolgoprudny, Russia;balashov73@mail.ru

Let H be the real Hilbert space. We shall say that a closed convex subsetA ⊂ H is R-strongly convex (R > 0) [1] if the set A can be represented asan intersection of closed balls of radius R. For a subset A ⊂ H we define thedistance function A(x) = inf

a∈A‖x − a‖ and the metric projection PAx = a ∈

A | ‖x− a‖ = A(x).

For a subset A ⊂ H we denote by UA(r) (r > 0) the neighborhood x ∈H | 0 < A(x) < r. We shall say that a closed subset A ⊂ H is proximallysmooth with constant R > 0 [2, 3] if the distance function A(x) is continu-ously differentiable on the set UA(R). The equivalent property for a proximallysmooth set A ⊂ H with constant R > 0 is that the set PAx is singleton andmetric projection continuous on x for any x ∈ UA(r) [2].

Theorem 1. [4] Let A ⊂ H be a closed convex subset. Then the followingproperties are equivalent:

1) the set A is R-strongly convex,

2) for any r > 0 and any points x0, x1 ∈ H\UA(r), ai = PAxi, i = 0, 1, thenext inequality holds

‖a0 − a1‖ ≤ R

R + r‖x0 − x1‖.

The properties 1), 2) from Theorem 1 characterize the real Hilbert space.More precise, if in arbitrary strictly convex reflexive Banach space with Frechetdifferentiable norm (on the unit sphere) properties 1) and 2) are equivalent forany R > 0 then the norm of the Banach space is equivalent to some Euclideannorm.

Theorem 2. [5] Suppose that for a closed subset A ⊂ H and r > 0 thereexists a constant C > 1 such that

‖a0 − a1‖ ≤ C‖x0 − x1‖, ∀x0, x1 ∈ UA(r), ai = PAxi, i = 0, 1. (1)

and C is the smallest possible Lipschitz constant in (1). Then R0 = R CC−1

isthe largest possible constant of proximal smoothness for the set A.

34

Let A ⊂ H be a proximally smooth subset with the largest possible constantof proximal smoothness R. Then the Lipschitz constant C = R

R−r in Formula(1) is the smallest possible.

We want to admit that if C = 1 in Formula (1), then the set A is convex.Indeed, by Theorem 2 we obtain that the set A is proximally smooth withany constant R > 0. Hence the metric projection on the set A is unique andcontinuous at any point x ∈ H\A. By the famous Vlasov’s result [6] the set Ais convex.

The author was supported by the Russian Foundation for Basic Research(project no. 13-01-00295-a).

References

1. E. S. Polovinkin. “Strongly convex analysis,” Sbornik: Mathematics, 187,No. 2, 259-286 (1996).

2. F.H. Clarke, R. J. Stern, P. R. Wolenski. “Proximal smoothness and lower–C2 property,” Journal of Convex Analysis, 2, No. 1-2, 117–144 (1995).

3. R. A. Poliquin, R. T. Rockafellar and L. Thibault. “Local differentiability ofdistance functions,” Trans. Amer. Math. Soc., 352, 5231–5249 (2000).

4. M. V. Balashov, M. O. Golubev. “ About the Lipschitz property of themetric projection in the Hilbert space”, Journal of Math. Anal. Appl., 394,No. 2, 545–551 (2012).

5. M. V. Balashov. “ Proximal smoothness of a set with the Lip-schitz metric projection”, Journal of Math. Anal. Appl.,http://dx.doi.org/10.1016/j.jmaa.2013.04.070.

6. L. P. Vlasov. “Chebyshev sets and approximately convex sets”, Mathemat-ical Notes, 2, No. 2, 600–605 (1967).

35

One approach to optimization of nonlinear controlledsystems

A.S. Buldaev1

1 Buryat State University, Ulan-Ude, Russia; buldaev@mail.ru

A new approach on the basis of differential-algebraic boundary value prob-lems to the nonlocal improvement of controls is considered. It is illustrated inthe framework of nonlinear optimal control problem with free right end

Φ(u) = ϕ(x(t1)) +

∫

T

F (x(t), u(t), t)dt→ minu∈V

, t ∈ T = [t0, t1], (1)

x(t) = f(x(t), u(t), t), x(t0) = x0, u(t) ∈ U, t ∈ T, (2)

x(t) = (x1(t), x2(t), ..., xn(t)) – state vector, u(t) = (u1(t), u2(t), ..., um(t)) –control vector. U ⊂ Rm – compact set. Initial position x0 and control inter-val T are given. Functions F (x, u, t), f(x, u, t) and its derivatives Fx(x, u, t),Fu(x, u, t), fx(x, u, t), fu(x, u, t) are continuous with respect to arguments (x, u, t)on the set Rn × U × T . Denote x(t, v), t ∈ T – solution of problem (2) for anyadmissible control v(t), t ∈ T .

We set the improvement problem of control u0 ∈ V with the respect to thefunctional (1): find the control v ∈ V with the following condition Φ(v) ≤ Φ(u0).

Let us solve the boundary value problem

x(t) = f(x(t), u∗(p, x, t), t), x(t0) = x0, (3)

p(t) = −Hx(p(t), x(t, u0), u0(t), t)− r(t), p(t1) = −ϕx(x(t1, u0))− q, (4)

〈Hx(p(t), x(t, u0), u0(t), t), x(t)− x(t, u0)〉+

+〈r(t), x(t)− x(t, u0)〉 == H(p(t), x(t), u0(t), t)−H(p(t), x(t, u0), u0(t), t),

(5)

〈ϕx(x(t1, u0), x(t1)− x(t1, u0)〉+ 〈q, x(t1)− x(t1, u

0)〉 == ϕx(x(t1))− ϕx(x(t1, u

0)),(6)

where the mapping u∗(p, x, t) = argmaxw∈U

H(p, x, w, t), H – Pontryagin’s func-

tion, the variable r(t) = (r1(t), r2(t), ..., rn(t)), t ∈ T , is determined at each timet ∈ T from the algebraic equation (5), q is satisfied to the algebraic equation(6).

Let us form the output control v by the next rule v(t) = u∗(p(t), x(t), t). Itis shown that in case of solubility of boundary value problem (3) – (6) admissibleoutput control v ∈ V provides the improvement of control u0 ∈ V .

36

If the control u0 satisfies the maximum principle, then in the case of non-uniqueness of the boundary value problem’s solution, the output control v canimprove this control u0.

Features of the proposed approach:1. Boundary value improvement problem is easier according to the smooth-

ness properties than the boundary value problem of the maximum principle.2. In case of linear optimal control problem with respect to state boundary

improvement problem split into two independent continuous Cauchy problem.3. The procedure allow to strictly improve any controls that do not satisfy

the maximum principle.4. The procedure have the opportunity to improve controls, satisfying the

maximum principle.5. The procedure allow us to formulate new necessary optimality conditions

that enhance the maximum principle in considered class of problems.6. Nonlocality of improvement.7. Absence of weak or needle variation control transaction.Given work is supported by the Russian Foundation for Basic Researches

(projects no. 12-01-00914, no. 12-01-98011). Theory and applications of themethod are published in articles.

References

1. A.S. Buldaev, O.V. Morzhin. “Improvement of controls in nonlinear systemson basis of boundary value problems,” Irkutsk State University Journal. Ser.Mathematics, Vol. 2, No. 1, 94–106 (2010).

2. A.S. Buldaev. “Boundary-value Improvement Problem for Linearly ControlledSystems,” Automation and Remote, No. 6, 87–94 (2011).

37

Optimal control of hydroelectric power stations

Vladimir Bushenkov1, Margarida Ferreira2, Ana Ribeiro3, GeorgiSmirnov4

1 University of Evora, Evora, Portugal; bushen@uevora.pt2 University of Porto, Porto, Portugal; mmf@fe.up.pt

3 University of Porto, Porto, Portugal;anafilipacastroribeiro@hotmail.com

4 University of Minho, Braga, Portugal; smirnov@math.uminho.pt

We study the problem of energy production optimization for a system ofhydroelectric power stations. The stations are equipped with turbines allowingnot only to produce the electric energy but also to pump water up from adownstream reservoir to an upstream one. A simplified model for these systemsis analysed in the context of optimal control theory with the fluxes of water toturbine or pump on each power station as control variables and the maximizationof the profit of energy sale being the objective function. The presence of stateconstraints and the nonconvexity of the cost function contribute to an increaseof complexity of the problem. Solving this problem amounts to maximizing anondefinite quadratic form subject to linear constraints in a Hilbert space. Wegive a theoretical analysis of this problem and deduce sufficient conditions ofoptimality.

After constructing a discretization of the problem, we numerically obtainits global solution using the method developed by Chen and Burer [1]. Twodifferent approaches are discussed and compared. In the first one, we directlyapply the Chen-Burer algorithm. In the second approach we use a specificstructure of the cost function that allows us to reduce the dimension of theproblem constructing a projection of the set of feasible solutions onto a subspaceof the cost function arguments. Then the Chen-Burer algorithm is applied tothe ”projected” low-dimensional problem. After that the solution to the originalproblem is obtained solving a simple convex programming problem.

References

1. J. Chen, S. Burer. “Globally solving nonconvex quadratic programming prob-lems via completely positive programming,” Math. Prog. Comp., 4, 33-52(2012).

38

Destruction of simple asymptotic trajectories ineconomic dynamical optimization models

Sergey Chukanov

Dorodnicyn Computing Centre of RAS, Moscow, Russia; chukanov47@mail.ru

Dynamic aspects of economic theory are often formalized as optimal con-trol problems and their generalizations. In the 60-80s of the last century, theturnpikes theory developed rapidly which is an analog to the theory of stabilityfor solutions of optimal economic dynamics. In terms of content, the resultsof this theory gave a simple representation of a ”systematic and proportional”development of the economy, and in the computational sense, a good initial ap-proximation. Later, it was found that, as the discount coefficient increases, theAndronov-Hopf bifurcation may generate a periodic regime in a neighborhood ofthe turnpike (steady state), which takes the role of an attractor (Benhabib andNishimura, 1979). Moreover, it was proved (Boldrin and Montrucchio, 1986),that the asymptotically strong discounting may generate an optimal synthesisdescribed by almost any (smooth) dynamical system, and thus by arbitrarilycomplex dynamics up to the chaotic. The scenario of evolution of the dynamiccomplexity is not still clear.

The studies of a number of applied optimal economic dynamical models,conducted in the Computational Center of RAS, show the lost of the turnpikestability; moreover, when approaching the boundary of stability domain, onefaces some multi-frequency weakly damping oscillations, a significant complica-tion of solutions, and a dramatic increase in computational complexity. Thesecircumstances motivated the present study. To analyze the development of thedynamic complexity, we use the approach based on the study of a sufficientlyrich parametric family of problems in which the solution of the Bellman equa-tion is a priori known. For numerical experiments, a software package in theMaple environment is developed that combines applied methods of the theoryof dynamical systems. It includes methods for calculating Lyapunov exponents,the correlation dimension of the attractor, the calculation procedure for thespectral characteristics of the dynamical variables and for visualization of theinvariant (natural) measures on the attractor. Some results of calculations ofdynamical systems of optimal synthesis, made according to the above scheme,will be presented and discussed.

This research was supported by the Program for Basic Research (PBR)number 15 of the Presidium of RAS, Project no. 106.

39

Optimization problem of the definition of dividendpayment

Dmitry Denisov1, Olga Nekrasova2

1 Lomonosov Moscow State University, Faculty of computational mathematicsand cybernetics, Moscow, Russian Federation; dvden@cmc.msu.ru

2 Lomonosov Moscow State University, Faculty of computational mathematicsand cybernetics, Moscow, Russian Federation; necros 0889@mail.ru

Modeling of various aspects of the financial activities of the insurance com-panies at the moment is an intensively developing area of mathematical researchwith many practical applications.

This paper considers the optimization control problem of the insurance com-pany with analysis of the discrete model, in which the s - initial capital, c -the flow of premiums in the unit of time under the signed contracts in thatcompany, Xt is a sequence of non-negative independent identically distributedrandom variables characterizing the aggregate amount of the loss for the t time.

Excluding dividend payments and the possibility of reinsurance, reserves ofthe company at the time t > 0 will be equal to R(t):

R(t+ 1) = R(t) + c−Xt+1, R(0) = s. (1)

In case of closure of t period with successful financial results in the beginningof next t + 1 period shareholders shall decide upon the strategy of dividendpayments. Consider d(t) is dividend payments at the beginning of t+ 1 periodfor the financial condition of the company at the time t, t > 0, then the formulafor finding the reserves of the company at time t+1 would be modified as follows

Rd(t+ 1) = Rd(t)− d(t) + c−Xt+1. (2)

With the need to improve the reliability of its financial activity the man-agers of the insurance company may adopt a decision to purchase reinsuranceprotection, thereby partially replacing the assumed obligations for reparationof loss of unknown quantity on the fixed cost of the reinsurance protection.Thus, reinsurance allows the assignor to assume the risks that are known tohave violated the balance of the portfolio as a whole.

With the ability to choose reinsurance strategy we get the following equationfor the reserves of the company with the remaining part of her risk of g(t) =g (Xt):

Rdre(t+ 1) = Rdre(t)− d(t) + cre − g (Xt+1) . (3)

40

We consider additional conditions on the probability of bankruptcy of thecompany with the final planning horizon Tfin. If τdre is the time of ruin, thenruin probability will be written in the following form:

ψdre(s) = Pτdre 6 Tfin

. (4)

Using the quantile criterion, we find the parameters of the model for a givenconfidence level 0 < α 6 1 of ruin probability : ψdre(s) 6 α.

The optimization problem for selection of strategies - reinsurance and div-idend payments, for which the total discounted value of dividends paid on theplanning horizon Tfin will be maximized subject to the restrictions on the prob-ability of ruin, will be:

T =

τdre, if τdre 6 Tfin;

Tfin, if τdre > Tfin, (5)

u(s, α) = E

[T−1∑

t=0

vtd(t)

]−→ max

d(t) ,g(t) , t∈[0,T ] , ψdre(s)6α , 0<α61

. (6)

In the recent years lots of actuarial articles were focused on the problemof selecting the optimal dividend strategy, maximizing of the expected presentvalue of dividend payments, see [1,2]. In [3] the classical continuum modelwas considered with an expression for the expected present value of dividendpayments for exponential loss distribution and the numerical examples.

References

1. H.U. Gerber, E.S.W. Shiu. “Optimal dividends: analysis with Brownian mo-tion,” North American Actuarial Journal, 1, No. 1, 1–20 (2004).

2. D.C.M. Dickson, H.R. Waters. “Some optimal dividends problems,” ASTINBulletin, 1, No. 34, 49–74 (2004).

3. D.C.M. Dickson, S. Drekic. “Optimal dividends under a ruin constraint,”Annals of Actuarial Science, 1, No. 1, 291–306 (2006).

41

Fast and slow dynamic feedbacks

V.V. Dikusar1, P.P. Gusyatnikova2

1 Computing Center RAS, Moscow, Russia; dikussar@ccas.ru2 Computing Center RAS, Moscow; pol@graffity.biz

In general case input-output system can be presented as:

x = f(x, u),y = ϕ(x),

which with introduced feedback isz = g(z, y),u = ψ(z),

where x ∈ Rn is state vector, y ∈ Rq is output vector, u ∈ Rp is input vectorand z ∈ Rm is feedback vector.

1. Fast dynamic feedbackConsider the fast feedback system

x = f(x, u),εz = r(z, y),

with y = ϕ(x) and u = ψ(z) .A s s u m p t i o n 1. 1. Functions f , ϕ , r and ψ are continuous.A s s u m p t i o n 1. 2. For every y the system z′ = r(z, y) has globally

asymptotically stable equilibrium z = φ(y). Function φ(y) is continuous.Define following static feedback, called the reduced system

x = f(x, u),z = φ(y),

with y = ϕ(x) and u = ψ(z).Theorem 1. Assume that system satisfies assumptions 1.1 and 1.2. Then

for every limited initial condition (x0, z0) , state vector x(t) of system is in-finitely close to the solution x0(t) of reduced system with initial condition x0 aslong as t and x0(t) remain limited and t ≥ 0.

1. Slow dynamic feedbackConsider the slow feedback system

x′ = f(x, u),z′ = εr(z, y)

42

with y = ϕ(x) and u = ψ(z), and where (.)′ =d(.)

dτ. We are interested in very

long time behavior of system. Thus introducing of time scaling t = ετ givesfollowing system

x =1

εf(x, u),

z = r(z, y)

with y = ϕ(x) and u = ψ(z).A s s u m p t i o n 2. 1. Functions f , ϕ , r and ψ are continuous.A s s u m p t i o n 2. 2. For every u the system x′ = f(x, u) has globally

asymptotically stable equilibrium x = φ(u). Function φ(u) is continuous.Now define reduced system

x = φ(u),z = r(z, y)

with y = ϕ(x) and u = ψ(z).Theorem 2. Assume that system satisfies assumptions 2.1 and 2.2. Let

(x0, z0) be a limited initial condition and z0(t) be the solution of reduced systemwith initial condition z0. Let x0(t) = φ(ψ(z0(t))) . Then state vector x(t) ofsystem is infinitely close to x0(t) as soon as t is not infinitesimal and as longas t and x0(t) remain limited.

As example we consider biological systems with fast and slow feedbacks.The authors were supported by the Russian Foundation for Basic Research

(project no. 11-07-00201).

References

1. N. Haritonenko, Yu. Yatsenko. Mathematical Modeling in Economics, Ecol-ogy and the Environment, Kluwer Academic Press, London (1999).

43

Optimal Control Problems with Constraints

V.V. Dikusar1, N.N. Olenev1, M. Wojtowicz2

1 Dorodnicyn Computing Centre of RAS, Moscow, Russia; dikussar@ccas.ru2 Technology and Humanismu University, Radom, Poland;

mar.wojtowicz@gmail.com

We consider the optimal control problems from applications in the aerospacefield, economics and financial mathrmatics. Expecially those problems are con-sidered which include control-state variable constraints [1,2]. An optimal controlproblem can be solved completely or at least qualitatively with the help of themaximum principle (MP). Difficulties in the numeric solving the optimal con-trol problems with control-state constraints (CSC) might derive from insufficientanalysis of model problems and inadequate progress in the MP theory in thecase. The MP inself could be a base of the numeric methods. Availability ofstate variable and CSC of a complex nature leads to a complication of the for-mulation and properties of the MP. New objects appear: measure and functionalLagrange multipliers. It becomes necessary to examine the properties of theseobjects and analyze the interconnections of the various parts of the MP. Oth-erwise it would be impossible to use the MP. It is known that the MP reducesthe initial control problem to the solution of the two-point (multi-point bound-ary value problems). They include: initial value problem (IVP); the problemof linear (LP) or nonlinear (NP) programming; root finding of transcendentalfunctions. The IVP consists of the two groups of nonlinear differential equations(direct and adjoint). A number of investigators have tried and rejected shootingmethods for certain classes of two-point problems which are particularly sensi-tive to the initial conditions and are thereby troublesome numerically. On theother hand there are successful examples of shooting methods at the expense ofa special parametrisation and continuation (homotory chains) the solution (R.Bulirsch, H.J. Pesch, J. Stoer, J. Bett, H. Maurer, A. Afanasjev, E. Smoljakovand others). One of the most effective numerical techniques for the solutionof optimal control problems is discretizing the differential equations. This ap-proach combines a nonlinear programming problems with discretization. Theresulting problem is characterized by matrices which are large and sparse. Theiteration routine requires an good initial quess for the adjoint variables. Suchtreatment of the problem has been succesfully utilized for applications. Thispaper extends the continuation methods for ill-posed problems. Our approachutilized the system dynamics and adjoint differential equations defined by themaximum principle. The resulting boundary value problem is characterizedby Jacobi or Hesse matrices which are small (dimension) and ill-conditioned.

44

Method of introduction the parameter helps to overcome the difficulties associ-ated with adjoint initial values.

Among the topics covered are:• Investigation of the degeneracy phenomenon of the maximum principle for

optimal control problems with state constraints and CSC.• Stiff ordinary differential equations.• Ill-posed linear and nonlinear programming problems.• Continuation method in ill-posed boundary valued problems.• Reentry body problem.• Optimal control problems in economics.• Banking activity problems.• Determination of geometry for optimal trajectory with inequality con-

straints.• Parallel calculations.

The work is supported by Russian Foundation of Basic Research (grant11-07-00201).

References

1. Afanasyev A.P. Dikusar V. V., Milyutin A.A., Chukanov S. A. Necessarycondition in optimum control. M: Nauka, 1990.

2. Dikusar V. V., Milyutin A.A. Qualitative and numerical methods in maxi-mum principle. M.: Nauka, 1989.

45

Algorithms for creation of adaptive grids fornon-convex unconstrained optimization

Anna Dorjieva1

1 Institute for System Dynamics and Control Theory of SB RAS, Irkutsk,Russia; ikadark@list.ru

We propose two conceptual approaches for design of algorithms for non-convex unconstrained minimization, based on adaptive grids [1]. Note thatadaptive grids prove themselves useful in minimization of expensive black-boxfunctions [2]. Consider the optimization problem f(x) → min, x ∈ D, D =x ∈ Rn : x < x < x. Here x, x ∈ Rn are given vectors.

Assume we are given a set D = xk : xk ∈ D, k = 1, ND of randomlychosen points. The first approach we propose is based on using the modified in-terpolation Sheppard function [3] as a so-called support function. The modifiedSheppard function FS(·;D) coincides with f(·) on D, and

FS(x;D) =

ND∑

k=1

fk + ak(x− xk)∥∥x− xk∥∥4

/ ND∑

k=1

1∥∥x− xk∥∥4

on D \ D. Here fk = f(xk), and ak ∈ R are coefficients of inclination angles ofthe function FS at points xk ∈ D.

The grid is constructed by the following recurrent process: set D0 := D,given Di we find a minimizer x∗ of the support function FS(·;Di), and setDi+1 := Di ∪ x∗. The second approach is based on a built-in mechanism forrough estimating of a Lipschitz constant combined with a special procedure ofelimination of uninformative points.

The author was supported by the RFBR (project no. 12-07-33021).

References

1. A.S. Lebedev, V.D. Liseikin, G.S.Khakimzyanov. “Development of methodsfor adaptive grids,” Computational Technologies, 7, No. 3, 31–37 (2002).

2. D.R. Jones, M. Schonlau, W.J. Welch . “Efficient Global Optimization ofExpensive Black-Box Functions, ” Journal of Global Optimization, 1, No.13, 455–492 (1998).

3. D. Shepard. “A two-dimensional interpolation function for irregularly-spaceddata,” Proceedings of the 23rd ACM national conference,517–524 (1968).

46

Optimal stabilization of multiply connected controlledsystems with using of homogeneous vector functions

Olga Druzhinina1, Natalia Petrova2

1 Dorodnicyn Computing Center of RAS, Russia; ovdruzh@mail.ru2 Dorodnicyn Computing Center of RAS, Russia; lyza@ccas.ru

In this work we suggest the conditions and algorithms of optimal stabiliza-tion for controlled dynamic systems described by nonlinear multiply connectedsystems of ordinary differential equations. We consider the common case andthe case when right-hand sides consists of homogeneous vector functions. Theproperties of stabilizing control and form of integrand in criteria of quality oftransient are used taking into account that subsystems are asymptotically sta-ble. Optimal control is synthesized at the level of the initial system.

For nonlinear dynamic systems methods of optimal stabilization are consid-ered in [1], [2] and other works. Critical cases are allocated and ways of a findingof stabilizing control in critical cases are developed [3]. For multiply connectedsystems the convenient way and the basis of general scheme of stabilization istwo-level stabilization [4].

It is known that for nonlinear systems of differential equations of generalform the conditions of theorems on asymptotic stability on the first nonlinearapproach are hardly verified. However, in some cases is possible to search forfairly easily verifiable conditions under which we prove the asymptotic stabilityof the equilibrium of the first nonlinear approach. Nonlinear system with right-hand sides are homogeneous (generalized homogeneous) vector functions havebeen studied in [5], which shows the theorems about asymptotic stability on thefirst nonlinear approach with the conditions which are easily verified.

With applications of results of works [6], [7] the algorithms of optimal sta-bilization for multiply connected nonlinear controlled systems are constructed.The results about optimal stabilization with respect to a part of variables aredeveloped with respect to a homogeneous case.

We consider multiply connected nonlinear controlled dynamic system

dxsdt

= X(µs)s (xs) +

q∑

j=1

Rsj(t, x) +Bs(xs)us ≡ Φs(t, x, u). (1)

Here xs ∈ Rns , x =(xT1 , ..., x

Tq

), X

(µs)s (xs) are homogeneous of order µs > 1

continuously differentiable vector functions, us ∈ Rrs , Rn1 ⊕ ... ⊕ Rnq = Rn,Rr1 ⊕ ...⊕Rrq = Rr, s = 1, q.

47

For the system (1) the problem of optimal stabilization has a unique solu-tion in closed form. According to Rumyantsev and Krasovsky theorems optimalcontrol and optimal Lyapunov function are obtained. The functional of controlJ(u0) is defined. To system (1) and to functional J(u0) we applied the resultsabout optimal stabilization of common nonhomogeneous multuply connectedsystem [7]. In the case under consideration we use general scheme of stabiliza-tion. The algorithms of optimal stabilization of system (1) with respect to alland to a part of variables are constructed.

The authors were suppoted by the Russian Foundation for Basic Reseach(project no. 13-08-00710-a).

References

1. V.V. Rumyantsev. ”On optimal stabilization of controlled systems”, Appl.Math. Mech., 34, No. 3, pp. 440–456 (1970).

2. N.N. Krasovsky. ”Stabilization problems of controlled motions”. In book:I.G. Malkin, Stability theory of motion. Nauka, Moscow, 1987, pp. 475–517.

3. G.M. Hitrov. ”To stabilization problem in critical cases”, Stability theory andits applications, Novosibirsk, Nauka, pp. 136–142 (1979).

4. D. Shil’yak. Decentralized control by complex systems, Mir, Moscow (1994).

5. A.Yu.Alexandrov. Stability of motions of nonautonomous dynamical systems,Saint-Petersburg, Saint-Petersburg State University (2004).

6. A.A. Shestakov, O.V. Druzhinina. ”Lyapunov function method for the analy-sis of dissipative autonomous dynamic processes”, Differential Equations, 45,No. 8, pp. 1108–1115 (2009).

7. O.V. Druzhinina, N.P. Petrova. ”On optimal stabilization with respect to apart of variables for multiply connected controlled systems”. Proc. of IIIInternational conference ”Optimization and applications” (OPTIMA-2012)held in Costa da Caparica, Portugal, in September 23 – 30, 2012. Universityof Evora, Dorodnicyn Computing Center of RAS, Moscow, 2012, pp. 67–71.

48

Primal-dual optimization methods for findingsaddle-points in differential games

Pavel Dvurechenskiy1, Yurii Nesterov2

1 Moscow Institute of Physics and Technology, Dolgoprudny, Russia;pavel.dvurechensky@gmail.com

2 Universite’ catholique de Louvain, Louvain-la-Neuve, Belgium;Yurii.Nesterov@uclouvain.be

Consider two moving objects with dynamics given by the following equa-tions:

x(t) = Ax(t)x(t) +B(t)u(t), x(0) = x0. (1)

y(t) = Ay(t)y(t) + C(t)v(t), y(0) = y0. (2)

Here x(t) ∈ Rn, y(t) ∈ R

m are the phase vectors of these objects, u(t) is thecontrol of the first object (pursuer), v(t) is the control of the second object(evader). Matrices Ax(t), Ay(t),B(t), C(t) are continuous and have appropriatesize. The system is considered on the time interval [0, θ]. Controls are restrictedin the following way u(t) ∈ P ⊂ R

p, v(t) ∈ Q ⊂ Rq ∀t ∈ [0, θ]. We assume

that P,Q are closed convex sets. The goal of pursuer is to minimize the valueof the functional:

∫ θ

0

F (τ, u(τ ), v(τ ))dτ + Φ(x(θ), y(θ)) = F (u, v) + Φ(x(θ), y(θ)). (3)

The goal of the evader is the opposite. We need to find optimal guaranteed resultfor each object which leads to the problem of finding the saddle-point of thefunctional above. We assume that u(·) ∈ L2([0, θ],R

p), v(·) ∈ L2([0, θ],Rq),

that the saddle point in this class of strategies exists, and that F (u, v) is uppersemi-continuous in v and lower semi-continuous in u and Φ(x, y) is continuous.

We consider two types of the functional (3)

1. F (u, v) and Φ(x, y) are convex-concave.

2. F (u, v) and Φ(x, y) are strongly convex-concave.

In each case we use Lagrange principle and add linear constraints (1), (2) tothe functinal. Then we perform a transition to the adjoint problem which turnsout to be finite-dimensional saddle-point problem of finding

minλ

maxµ

ψ(λ, µ), (4)

where λ,µ are Lagrange multipliers.

49

In the first case we find partial subdifferential and superdifferential forψ(λ, µ) in explicit way an prove that they are bounded. Next we use the methodof simple dual averages (SDA) introduced in [1] to find approximate solutionfor (4). After that we find approximate solution for initial problem using thesequence generated by SDA method and prove that it is nearly optimal and

nearly feasible with the error O(

1√k

), where k is the number of iteration.

In the second case we prove that the function ψ(λ,µ) is smooth and find itspartial gradients in explicit form. We show that the problem (4) can be reducedto a problem of finding a weak solution of a variational inequality

find x∗ s.t. 〈g(x), x− x∗〉 ≥ ∀x ∈ S,

where the operator g(x) is Lipschitz continuous and monotone. We use themethod introduced in [2] to find approximate solution for (4). After that wefind approximate solution for initial problem using the sequence generated thismethod and prove that it is nearly optimal and nearly feasible with the errorO(1k

), where k is the number of iteration.

This research is supported by grant from the government of Russian Feder-ation (contract 11.G34.31.0073)

References

1. Yu. Nesterov “Primal-dual subgradient methods for convex problems,” Math-ematical Programming., 120, No. 1, 221–259 (2009).

2. Yu. Nesterov “Dual extrapolation and its applications for solving variationalinequalities and related problems,” Journal Mathematical Programming: Se-ries B., 109, No. 2, 319–344 (2007).

50

Subsystems of Affine Control Systems

Vladimir Elkin1

1 Institution of Russian Academy of SciencesDorodnicyn Computing Centre of RAS , Moscow, Russia; elk v@mail.ru

Affine control systems are nonlinear systems that are linear in controls; that,is, these are the systems of the form

y = f0(y) + f(y)u, y ∈M ⊂ Rn, u ∈ Rr. (1)

Here, y are the phase variables; u are the controls; and M is the phase spacethat is a domain. We assume that f0 is a smooth vector field; f is an n-by-rmatrix in which the columns fα, α = 1, 2, . . . , r, are smooth vector fields; andrankf(y) = const.

Let N ⊂ M be an elementary m-dimensional manifold, i.e. a set that isdiffeomorphic to a domain in Rm. Thus, N is the image of a domain L ⊂ Rm

under a smooth mapping χ : L→ Rn.A subsystem is the restriction of system (1) to the manifold N belonging to

the phase space M . More precisely, it is a system

x = g0(x) + g(x)v, x ∈ L ⊂ Rm, v ∈ Rs, (2)

which has the following property. For any solution x(t) to system (2), y(t) =χ(x(t)) is a solution to system (1).

The need for restricting a control system (1), i.e., changing over to a sub-system (2) on a subset N ⊂M , is dictated by practical considerations when theelements of the set M are forced to obey certain constraints (initial conditions,boundary conditions, etc.). In such cases, it is natural to restrict a system (1)to some subset N ⊂M for which these constraints are satisfied. The subsystem(2) on N defines a part of the solutions of the initial system S that lie withinN and, in particular, satisfy the given constraints. Therefore, the solution ofa control problem formulated for a system (1) can be reduced to the solutionof an analogous problem for its subsystem (2) with phase space of diminisheddimension.

The authors were supported by the Russian Foundation for Basic Research(project no. 13-01-00866).

51

A 2-approximation algorithm for finding the cliquewith minimum weight of vertices and edges

Ivan Eremin1, Edward Gimadi2, Alexander Kelmanov3, ArtemPyatkin4, Mikhail Khachay5

1,5 Institute of Matematics and Mekhanics UroRAN, Ekaterinburg, Russia;eremin@imm.uran.ru; mkhachay@imm.uran.ru

2,3,4 Sobolev Institute of Mathematics SB RAS, Novosibirsk, Russia;gimadi@math.nsc.ru; kelm@math.nsc.ru; artem@math.nsc.ru

One of the most important data analysis problems — the cluster analysisproblem — is considered. Its essence is in partitioning the set of n objects intogroups of similar ones.

Frequently the input for the problem is given as a matrix of pairwise com-parisons of objects (the values of numerical characteristics are not available),and the criteria for comparison can be various. In this paper we consider ageneralization of this problem for the case when the objects are weighted.

Let the input of the problem consists of the weight function a = (ai) ofobjects and the matrix c = (cij), 1 ≤ i, j ≤ n, of pairwise comparisons of theobjects, where ai and cij are real numbers. The aim is to find a subset of objects(rows of the matrix) with the minimum (maximum) sum of their weights andthe weights of pairwise comparisons of these objects. This problem can bereformulated as follows.

Weighted Clique Problem (WCP).G i v e n a complete weighted undirected graph G = (V,E, a, c), where a :

V → Q, c : E → Q and a natural number m < n.F i n d in the graph G a clique C of order m with the lowest (highest) total

weight of objects (vertices and edges) belonging to this clique.

Claim 1. The problem WCP in terms of the modified weights of edges

wij =

aim−1

+ cij +ajm−1

, if i 6= j;

0, if i = j,is written in the equivalent form

∑

e∈E(C)

we → minC⊂V ;|C|=m

(max

C⊂V ;|C|=m

),

where E(C) is set of edges in the m-clique C.

52

Claim 2. The problem min-WCP cannot be solved by a polynomial approx-imation algorithm with an arbitrary fixed guarantee unless P = NP.

Remark. In fact, for the problem min-WCP even a polynomial O(2n)-approximation algorithm (where n = |V |) cannot exist unless P = NP.

We now return to the previously mentioned data analysis problem with an× n matrix of pairwise comparisons of the objects as an input.

Problem RSSM (Row’s Subset of Symmetric Matrix).I n p u t: A symmetric n × n-matrix (wij) with non-negative elements wherewii = 0, a natural number m < n and a positive D.Q u e s t i o n: Is there a subset C of m rows of this matrix such that

∑i∈C∑j∈C wij ≤

D.It is easy to see that the problem RSSM is polynomially equivalent to the

problem min-WCP formulated in the form of properties verification.

Theorem 1. The problem min-WCP is strongly NP-hard.

We use an exact solution of the problem of finding minimum-weight m-star in the graph G = (V,E, a, c) as an approximate solution of the problemmin-WCP [1]. This star can be found evidently in a time O(n2).

Theorem 2. Let the elements of (cij) satisfy the triangle inequality cij +cjk ≥ cik for any i, j, k. Then the problem min-WCP on G can be solved in timeO(n2) with asymptotically attainable performance ratio 2.

Theorem 3. Let the elements of (wij) are computed as in Claim 1 wherethe matrix (cij) consists of the squares of pairwise distances within a system ofpoints in Euclidean space. Then the problem min-WCP on the graph G is solvedin time O(n2) with achievable performance ratio 2.

The authors were supported by the RFBR (projects 12-01-00090, 12-01-00093, 12-01-33028, 13-01-00210, 13-07-00070, and 13-07-00181) and Targetprograms of UB RAS and SB RAS (integration projects 01.12.1017/1 and 7B).

References

1. I. I. Eremin, E.Kh.Gimadi, A. V.Kelmanov, A.V. Pyatkin, M.Yu.Kha-chay. “A 2-approximation algorithm for finding a click with minimum weightof vertices and edges”, Proceedings of the Institute of Mathematics and Me-chanics, Ural Branch of RAS, 19, No. 2, (2013) (in Russian, to appear).

53

Wireless sensor networks and covering a plane withsectors

Adil Erzin1

1 Sobolev Institute of Mathematics, Novosibirsk, Russia;adilerzin@math.nsc.ru

The set C of plane figures is a cover of the plane region R of area S if eachpoint in R belongs to at least one figure in C. The density of the plane region’scover is the ratio of the sum of the element’s areas in the cover C to the region’sarea S. The efficient cover is one with minimal number of equal plane figures(elements of the cover) per area.

In wireless sensor networks (WSN), the sensing area of each sensor may bea plane figure (usually disk, ellipse or sector), and it is said that the sensorcovers the figure. In WSNs, each sensor is supplied with a limited amount ofnon-renewable energy, and its lifetime is inversely proportional to the coveredarea. In order to prolong the WSN’s lifetime one can minimize the density of acover.

The most studied covers are regular covers, where a planar region is tiledby identical regular polygons (tiles), and all of the tiles are covered equally. Inthis case, the density of the planar cover is defined by the coverage density ofone tile. In [1], we introduced the classification of regular covers, according towhich cover in class COVk(p, q) covers each regular k-angle polygon by p figuresof q different types.

In this paper, we solved two problems:(1) construction of regular plane covers with a minimal density in the classesCOV3(p, 1) when a tile is an equilateral triangle and one tile is covered by p,p = 1, 2, 3, equal sectors;(2) construction of efficient regular plane covers using equal sectors.

The author was supported by the Russian Foundation for Basic Research(project no. 13-07-00139).

References

1. V. Zalyubovskiy, A. Erzin, S. Astrakov, H. Choo. “Energy-efficient area cov-erage by sensors with adjustable ranges”, Sensors, No. 9, 2446–2469, (2009).

54

Parallel computations in processes of formalizationprinciples of the method of branches and borders and

solving combinatorial tasks

Georgy Evdonov1

1 National Research Nuclear University ”MEPhI”, Moscow, Russia;evdonovg@gmail.com

It is well known that the method of branches and borders in general formu-lation has several informal principles, each of them can be specified dependingfrom concrete type of task. Currently most implementations of the method ofbranches and borders consider particular type of problem, so for different tasksthe calculation of subset borders is different. Thus, the developer must programa new algorithm to solve a new type of task.

The goal of this work is to deliver developer from these actions by real-ization syntax constructions for entering corresponding data.The main task forimplementation of the proposed approach is to create a procedure to determinethe calculation mechanism of a set border and the procedure to split a set.

For implementation procedures of the bound calculation it is nesessary to:

1. Express bound in a mathematical form;

2. Convert this form in a syntax of the software product;

3. Interpret this formula in procedures and functions of the software product;

4. Interpret the parameters of the formula, and link them with the initialconditions;

5. Define a rule that determines a set.

During the parsing of the input formula arises the problem of the parametersidentification. It lies in the fact that it is unknown what parameters will beintroduced and what they mean. To avoid this problem the user must interpretparameters in separate section also. If parameters are calculated values, thensystem must verify that it can be presentd as superposition of functions, andall other data are atomic parameters which are contained in task condition.

The basic concept, which operates the principle of branching is ”a set”.Thus, for different types of tasks this term will have different definitions. Re-spectively for implementation of this principle it is necessary to allow the userto specify a set. To realize it system should have corresponding functionalityfor work with sets.

This approach has several disadvantages, such as complexity of the syntaxof input data and increasing processing time due to splitting input instructions

55

into a simple operators. The consistent implementation of this mechanism incomparison with the implementation of method of branches and borders for aprivate tasks will have a lower productivity. However, it is possible to acceleratecalculations through the usage specialized functions and parallel informationprocessing. To solve this problem it is offered to break the process of parsinginto a multiple threads where a separate thread will disassemble separate setsof input data.

After interpretation of initial data in the internal parameters of the system,along with the standard parallelization of branching, it is possible to use parallelcomputing for calculation of the border and substitution of relevant parameters.Such opportunity is provided by the fact that the estimation formula is a su-perposition of a many atomic operators. Some of them are nested and thereforecannot be parallelized, but the remaining part can be calculated separately.

The main advantage of this method is that for programmer is not requiredto program a separate method of branches and borders for solving a particulartype of task. The main task of the researcher is to present the algorithm innotation of functions which are presented in a software package, and improve-ment of mechanisms for processing of the input data reduces the problem of acomputational efficiency.

References

1. V.P. Gergel. Theory and practice of parallel computing, BINOM KnowledgeLaboratory, Moscow (2010).

2. R.G. Strongin., Ya. D. Sergeev Global optimization with non-convex con-straints: Sequential and parallel algorithms, Kluwer Academic Publisher,Dordrecht (2000).

56

Towards Deterministic Constrained MultiobjectiveOptimization

Yury Evtushenko1, Mikhail Posypkin2

1 Dorodnicyn Computing Centre of RAS, Moscow, Russia; evt@ccas.ru2 Institute for Information Transmission Problems, Moscow, Russia;

mposypkin@gmail.com

The talk describes a method for solving multi-objective optimization prob-lems with non-linear functional constraints:

minx∈X

F (x), where X = x ∈ B : G(x) ≤ 0k. (1)

where F (x) : Rn → Rm defines criteria, G(x) : Rn → R

k defines constraints andB is an n-dimensional box. Mappings F (·), G(·) are continuous. Relation ≤ isdefined elementwisely and 0k, 1k denote k-dimensional vectors with all zero orunit components respectively.

Unlike existing approaches the proposed method not only generates a finiteapproximation of Pareto frontier but also guarantees its optimality to a givenaccuracy. The method constructs ε, δ-Pareto set, defined as follows. For a δ ≥ 0define Xδ = x ∈ B : G(x) ≤ δ · 1k and Y δ = F (Xδ). A set Q ⊆ Y δ is calledε, δ-Pareto set if

1. there are no such two points a, b in Q that a ≤ b and a 6= b;

2. for any y ∈ Y = F (X) there is a point q in Q such that q − ε · 1m ≤ y.

When δ = 0, ε, δ-Pareto set is simply ε-Pareto set introduced in [1]. We ex-tended the original Non-Uniform Covering Method proposed for box-constrainedproblems to a case of non-linear constraints. A practically important applicationof the proposed method to finding the working space of a robotic manipulatorwith constraints on end-effector angles is presented.

The authors were supported by the Russian Foundation for Basic Research(project no. 11-01-00786).

References

1. Y. Evtushenko and M. Potapov “ Non-differentiable approach to multicriteriaoptimization,” in: Nondifferentiable Optimization, Motivations and Applica-tions, 1983, pp. 43–45.

57

Functions associated with 3-SAT problem.

Rashid Faizullin1

1 OmSTU, Omsk, Russia; r.t.faizullin@mail.ru

Let it is given a Boolean function defined on the set of vectors y = [y1, . . . , yN ] ∈∈ false, trueN , presented in the conjunctive-normal form (CNF):

L∗(y) =M∏

i=1

G∗i (y),

where G∗i (y) =

∑Nj=1 q

∗i,j(y),

q∗i,j(y) =

yj , if yj included iclauseyj , if yj iclause with negation,

false, otherwise.

Form L∗ with the function F : [0, 1]N −→ R+, defined by the relationF (x) =

∑Mi=1 Ci(x), where Ci(x) =

∏Nj=1 pi,j(x) and

pi,j(x) =

x2j , ifq∗i,j(y) = yj ,

(1− xj)2, ifq∗i,j(y) = yj ,

1, otherwise.

(1)

By construction, the form L∗ has a satisfying set of truth y∗ (Ie, the cor-responding statement of the problem is solvable SAT) if and only if minF (x)is reached and zero, which is equivalent to a minimum point x∗ conditionsCi(x

∗) = 0, (i = 1, . . . ,M). Obviously, for each a set of truth y∗ formL∗ corresponds to the point x∗ minimum of F , the coordinates of which aregiven. Suppose further that L∗ satisfiable 3-CNF with N variables and Mclauses. Let’s assign to each of its disjuncts G∗

i set J ′i = j1, j2, j3 low

indexes of its member variables. After fixing the pair (y, y∗) where y∗ setof true literals, we define the index set of clauses by the following formulas:IF,k(y, y

∗) = i ∈ IF (y) : |j ∈ J ′i : y

∗j 6= yj| = k (k ∈ 1, 2, 3) We also

introduce set: ΦF,1 ⊆ IF,1 : ∀j ∈ J ′i ⇒ yj = false We describe the properties

of the function, which is the restriction of F on the interval [x, x∗], where thevectors x and x∗ obtained from the y and y∗ formulas . We define a function fequality F (t) = F ((1− t)x+ tx∗).

Proposition: 1. Let |ΦF,1(y, y∗)| ≥M/ 9, then f — monotonically decreas-ing function. 2. If we have the condition |ΦF,1(y, y∗)| ≥M/3, then f is convexfrom below on [0, 1].

58

An application of Chebyshev-Markov-Krein algorithmto the numerical solution of the control problem,described by the Smoluchowski kinetic equation

Zuhura Gaeva1, Alexander Shananin2

1 Alpine Geophysical Institute, Moscow, Russia; gaevaz@yandex.ru2 MIPT, CCAS, Moscow, Russia; alexshan@yandex.ru

We propose an approach to the numerical solution of the problem of controland management hail cloud. The evolution of the cloud is described by a systemof integro differential Smoluchowski equations for the distribution functions ofdroplets and crystals by weight and size. The main objective of management isto prevent the creation of large hailstones in the cloud that can cause damage.Thus the control problem is to minimize the number of hailstones in the cloudwhose size exceeds the critical size. Presently even the calculation of integralcharacteristics of the particle distribution function in the cloud, described by asystem of kinetic equations is challenging, not to mention the task of manage-ment. So the real issue is to develop effective methods to solve these equationswith the necessary accuracy.

To manage the problem we use the Galerkin approximation, which allows toreduce the control problem of a complex system governed by integro-differentialequations to the task of solving a system of ordinary differential equations.Moreover, we can significantly reduce the amount of computation by a specialchoice of the basis for the complete orthonormal system of basic functions. Theusage of the Galerkin method implies the issue of the accuracy of computationsas the integral characteristics that describe the evolution of a cloud cannot becalculated exactly from the information on the finite number of the Galerkin’scoefficients.

To resolve this issue we use the variable change to obtain the two-sidedbounds of the cumulative distribution functions of the integral characteristicvia the reduction to the Chebyshev-Markov-Krein momentum problem with anuncertain first moment. Thus, the difficult problem of controlling the distribu-tion of the hailstones in the cloud can be replaced by an optimal control task,described by a system of the ordinary differential equations. Though the func-tional of the later system upperbounds the functional of the former system it isnow not given explicitly. Its particular values can be obtained via the solutionthe corresponding Chebyshev-Markov-Krein momentum problem. On the otherhand, to draw the machinery related to the Pontryagin maximum principle, oneshould manage to compute not only the values of the functional, but its deriva-tives as well (e.g. to use numerical counterparts of the transversality conditions).

59

But point wise computation of the derivatives is highly unstable process. Toovercome this difficulty we manage to modify the Chebyshev-Markov-Krein al-gorithm to produce not only the values of the functional at points but for itsderivatives as well, without significantly increasing the computational complex-ity of the algorithm. Our approach enables to solve the control problem of themicrophysical processes in the hail cloud by applying vast variety of the numer-ical methods for the solution of the optimal control problems. Some specifics ofthe arising optimal control problem will also be discussed.

The authors were supported by RNP.2.2.1.1.2467, Department of Mathe-matical Sciences of RAS No.3, Presidium of RAS No.14, project 109.

References

1. Z.S.Gaeva, A.A. Shananin “The moment problem of Markov-Tchebishev andinvestigation of Galerkins approximation for some model of cristal aggrega-tion.” (in Russian), Mathematic Modeling, 7, No 9, (1995).

2. Z.S.Gaeva, A.A. Shananin. “Numerical method for the problem control microphysical process in clouds.” (in Russian), Mathematic Modeling, 16, No 12,(2004).

3. Z.S.Gaeva, A.A. Shananin. “Tchebishev- Markov-Krein‘s algorithms for anal-yses of micro physics process in heavy clouds.”, Journal of mathematicalphysics and computational mathematics, 47, No 9, (2007).

60

A 2-approximation algorithm for a NP-hard problemof searching a family of disjoint vector subsets

Alexander Galashov1, Alexander Kel’manov2

1 Novosibirsk State University, Novosibirsk, Russia;galashov.alexandr@gmail.com

2 Sobolev Institute of Mathematics, Novosibirsk, Russia; kelm@math.nsc.ru

We consider the following intractable problem.

Problem FDVS-(Rq)SD (Family of Disjoint Vector Subsets, Euclideancase for Squared Distances).

Given a set Y = y1, . . . , yN of vectors from Rq and positive integer num-

bers Mj , j = 1, . . . , J .Find a family of disjoint vector subsets Cj ⊆ Y, j = 1, . . . , J , such that

J∑

j=1

∑

y∈Cj

‖y − y(Cj)‖2 → min,

where y(Cj) = 1|Cj|

∑y∈Cj

y, under constraints |Cj | =Mj , j = 1, . . . , J .

This problem is strongly NP-hard even if J = 1 [1].

In this work we present a 2-approximation polynomial algorithm for the casewhen J is fixed. This algorithm runs in time O(N2(q +NJ+1)).

The authors were supported by the Russian Foundation for Basic Research(projects no. 12-01-00090, no. 13-07-00070 and no. 12-01-33028) and by theTarget Programme of the Siberian Branch of the Russian Academy of Sciences(integration projects no. 7B and no. 21A).

References

1. A.V. Kel’manov, A.V. Pyatkin. “NP-Completeness of Some Problems ofChoosing a Vector Subset,” Journal of Applied and Industrial Mathematics,5, No. 3, 352–357 (2011).

61

Optimization of packing equal ellipses into arectangular

Shamil Galiev1, Maria Lisafina2

1 Tupolev Kazan National Research Technical University; Kazan, Russia;sh.galiev@mail.ru

2 Tupolev Kazan National Research Technical University; Kazan, Russia;e marie@mail.ru

The linear models for the approximate solution of the problem of packingthe maximum number of equal ellipses of the given sizes into a rectangular Rare proposed. A heuristic algorithm for solving the packing problems based onlinear models is developed. Numerical results demonstrate the effectiveness ofthis approach.

As indicated by Hifi and M’Hallah, cutting and packing problems are scien-tifically challenging problems with a wide spectrum of applications. They arevery interesting NP-hard combinatorial optimization problems [1]. Presently,various models and methods of solving of packing problems are available thatoften reduce the packing problem to combinatorial problems or to nonlinearprogramming problems and so on, see [1-4].

In this paper, we propose 0-1 linear models for determining the maximumnumber of equal ellipses of the given sizes that can be packed into a givenrectangular R. We consider the packing ellipses when: (1) major axes of thepacking ellipses are parallel only one of the axes either x or y, (2) the majoraxes of some ellipses are parallel to the x axis, while others - y axis. We showthe acceptability of using the lp metrics to determine the conditions of non-intersection of ellipses with mutually perpendicular major axes.

We construct a grid in R; the nodes of this grid form a finite set of pointsT , and it is assumed that the centers of ellipses to be packed can be placedonly at the points of T . The packing problems of equal ellipses with the centersat the points of T are reduced to 0-1 linear programming problems. To solvethese linear programming problems of large dimensions a heuristic algorithm isproposed.

We have introduced the levels of the centers of the ellipses and weights ofthe levels. By selection of weights for different levels, we can ensure that theellipse centers are at the desired level. If the weights of levels are increasingfrom up to down, the optimization process is carried out under the pressingof the ellipses down. The packing result may depend on directions of ellipsespressing up or down, right or left.

62

As given rectangular, we selected unit square Q and rectangle R formedby two squares Q put one on the top of the other. To solve the problem ofpacking into the given domain a computer program based on CPLEX-11.2 wasdeveloped. For the evaluation of the ellipse packing quality we proceeded asfollows. We chose radius (r) of the circles and received packing of maximumnumber of equal circles of radius r into the square Q. Then we compared theresulting packing with the best known packing of circles of radius r in the unitsquare. Eccentricity of the ellipse was chosen, and the values of the major andminor axes of the ellipse are calculated so that the area of the ellipse coincidewith the area of the circle of radius r. Then we could compare the packingdensity of ellipses and circles. The obtained packings of circles in a square Qwith a few exceptions are the same as the best-known packings, see [2]. Thenumber of ellipses packed in Q, as a rule, is less than the number of circles(areas of which coincide with the areas of the ellipses). In the examples underconsideration when we pack ellipses in a rectangle R, it turned out that in thehalf cases the number of the packed ellipses and circles are the same.

The report contains obtained packings into the rectangular R up to 50 andmore ellipses for each of the above indicated cases (1) - (2). Numerical resultsdemonstrate the effectiveness of this approach.

References

1. M. Hifi and R. M’Hallah “A Literature Review on Circle and Sphere PackingProblems: Models and Methodologies,” Advances in Operations Research,Article ID 150624, 22 pages, doi: 10.1155/2009/150624 (2009).

2. P.G. Szabo, M. Cs. Marcot, T. Csendes, E. Specht, L.G. Casado, I.Garcia New Approaches to Circle Packing in a Square. With Program Codes,Springer, (2007).

3. L. F. Toth. “Packing of ellipses with continuously distributed area,” Journalof Discrete Mathematics, No. 60, 263–267 (1986).

4. W. X. Xu, H.S. Chen, Z. Lu “An overlapping detection algorithm for randomsequential packing of elliptical particles,” Physica A, No. 390, 2452-67 (2011).

63

Entropy optimization problem and new efficient dualmethod based on the regularization of dual problem

Alexander Gasnikov1, Yurii Nesterov2

1 Premolab MIPT, Dolgoprudny, Russia; avgasnikov@gmail.com2 CORE/INMA UCL, Louvain le Neuve, Belgium;

yurii.nesterov@uclouvain.be

In different applications we often have to solve the following entropy-linearprogramming problem:

f(x) =∑ni=1 xi ln xi +

∑ni=1 cixi → min

x∈QQ = x ≥ 0 :

∑ni=1 xi = 1

∑ni=1 aijxi = bj , j = 1, . . . ,m

(1)

Let’s introduce Lagrange function:

L(x, λ) =n∑

i=1

xi ln xi +n∑

i=1

cixi +m∑

j=1

(λj

n∑

i=1

aijxi − bj

). (2)

Consider the dual problem:

minx∈Q

maxλ

L(x, λ) = maxλ

minx∈Q

L(x, λ).

φ(λ) =m∑

j=1

λjbj + ln

(n∑

i=1

exp(−ci − 1−m∑

j=1

λjaij)

)→ max

λ(3)

It’s easy to see, that if we know the dual variables λ, we can calculate theprimal variables x (because of the explicit formulas for x(λ) = exp(−ci − 1 −∑m

j=1 λjaij)/(∑n

i=1 exp(−ci − 1−∑mj=1 λjaij)

)).

We choose 1-norm in x-space and 2-norm in λ-space. Our purpose is to findsuch λ that fo fixed ǫf , ǫg > 0 :

‖∇φ(λ∗)‖2 ≤ ǫg , 〈∇φ(λ∗), λ∗〉 ≥ −ǫf . (4)

If (4) is true, than:

x(λ∗) ∈ Q, ‖Ax(λ∗)− b‖2 ≤ ǫg , f(x(λ∗)) ≤ f∗ + ǫf , (5)

where f∗ = minx∈Q

f(x).

64

Consider a regularized problem (δ = ǫg/√lnn):

φδ(λ) = φ(λ)− δ/2‖λ‖22.The lipschitz constant of the gradient of φδ: Lφδ

= Lφ + δ.According to PhD thesis of Y.E. Nesterov (1983) it is worth to use FGM

method for maximization of φδ:

λk+1 = uk +∇φδ(uk)/(Lφ + δ)uk+1 = λk+1 + τ (λk+1 − λk), k ≥ 0,

(6)

where τ = ([Lφ + δ]1/2 − δ1/2)/([Lφ + δ]1/2 + δ1/2) and λ0 = 0.

The number of iteration for achieving the required accuracy:

O

(√Lφ∆

ǫgln

Lφ∆

ǫf ǫ2g

), (7)

where

∆ = minx∈Q

f(x) : Ax = b − minx∈Q

f(x) = φ∗ − φ(0) = ln(n)

Lφ =1

σ(f)‖A‖2 = max〈Ax,λ〉 : ‖x‖1 ≤ 1, ‖λ‖2 ≤ 1 = max

1≤i≤n‖ai‖2.

We use the fact that the constant of strictly convexity σ(f) of entropy functionf(x) in 1-norm is equal to 1.

The authors is partially supported by Laboratory for Structural Methodsof Data Analysis in Predictive Modeling, MIPT, RF government grant, ag.11.G34.31.0073 and by RFBR 11-01-00494-a, 12-01-33007 (program of support-ing of leading youth collectives), grant of President RF no MK-5285.2013.9.

65

Piecewise quadratic functions in the methods oflinear optimization

Alexander Golikov1

1 Dorodnicyn Computing Centre of RAS, Moscow, Russia; gol-a@yandex.ru

The report considers the issue of piecewise quadratic functions applicationin linear optimization methods. The functions concerned can be used in suchfields as finding normal solutions to linear equation systems and/or inequalities(using the theorems of alternatives), constructing hyperplanes family separatingpolyhedra, determining the steepest descent for convex programming problemsmethods, svm-recognition methods, determining the projection of a given pointon the set of solutions of linear programming (LP), etc. The piecewise quadraticfunction effectiveness is chiefly concerned with the generalized Newton methodapplied for their unconstrained minimization and duality theory.

Several parallel versions of the generalized Newton method were imple-mented for unconstrained minimization of convex piecewise quadratic functionsbased on various optimization schemes of the original linear problem’s datadecomposition (column, row, and cellular schemes). The resulting parallel algo-rithms were successfully used to solve large-scale linear optimization problems(up to several dozen millions of variables and several hundred thousands of lin-ear equations). Some computational experiments were performed on a clusterconsisting of two-processor nodes based on 1.6 GHz Intel Itanium 2 processorsconnected by Myrinet 2000. For example, a cellular scheme integrating 144processors within a cluster applied to solve an LP problem with one millionvariables and 10000 constraints accelerated the computation rate by approxi-mately 50 times, with computation time being as small as 28 s. LP problemwith two million variables and 200000 constraints was solved on 80 processorsin about 40 min. Another LP problem with 60 million variables and 4 thousandconstraints was solved on 128 processors by means of a column scheme in 140s.

The author’s work was supported by the Programs of Fundamental Researchof Russian Academy of Sciences P-18, by the Russian Foundation for BasicResearch (project no.11-01-00786), and by the Leading Scientific Schools Grantno.5264.2012.1.

66

A Numerical Method for Solving Bimatrix Games

Evgeny Golshteyn1, Ustav Malkov2, Nikolay Sokolov3

1 CEMI RAS, Moscow, Russia; golshtn@cemi.rssi.ru2 CEMI RAS, Moscow, Russia; ustav-malkov@yandex.ru3 CEMI RAS, Moscow, Russia; sokolov nick@rambler.ru

Let Γ be a bimatrix game (i.e., a finite two person game with mixed strate-

gies) defined by m × n-matrices A1 = (a(1)ij ) and A2 = (a

(2)ij ). Next, let Xi be

the set of mixed strategies of player i, i = 1, 2. On the set X = X1×X2 , definethe Nash function

N(x) = max1≤i≤m

n∑

j=1

a(1)ij xj2 + max

1≤j≤n

m∑

i=1

a(2)ij xi1 − x1(A1 + A2)x

τ2 ;

where N(x) ≥ 0 for all x ∈ X, and N(x) = 0 ⇔ x ∈ X∗. Therefore, the solu-tion of game Γ turns out to be equivalent to the search for the points of globalminimum (equal to zero) of the function N(x) over the set X. The latter prob-

lem is equivalent to the minimization of the Nash function N(x1, x2, v1, v2) =(v1 − x1A2x

τ2) + (v2 − x1A1x

τ2), where v1 and v2 are the scalar variables re-

lated to the vector-variables x1 ∈ X1 and x2 ∈ X2 by means of the constraints∑nj=1 a

(1)ij xj2 ≤ v2, 1 ≤ i ≤ m,

∑mi=1 a

(2)ij xi1 ≤ v1, 1 ≤ j ≤ n.

Having fixed one of the pure strategies of a player, we can find a localminimum of the Nash function by solving several pairs of the appropriate linearprogramming problems. After having tried all the pure strategies, we yieldeither an exact solution of the game Γ, or its approximate solution providingfor a small enough value of the Nash function.

The algorithm has been tested on a collection of 5000 randomly generatedgames with the dimensions m,n varying between 20 and 1000. Details of thenumerical results are presented in the talk. The experiments have demonstratedthe method to solve all the games such that their randomly generated matricesare filled completely. If the percentage of filled entries decreases, then somegames can be solved by the algorithm with some accuracy only; and the lessthe percentage is, the more the number of games solved approximately.

67

Local strong convexity in the Hilbert space

Maxim Golubev1

1 Moscow Institute of Physics and Technology, Dolgoprudny, Russia;maksimkane@mail.ru

Let H be a real Hilbert space. We denote by BR(x) = y ∈ H : ‖y−x‖ ≤ Rthe closed ball of radius R ≥ 0 and center x ∈ H . Let ∂A be the boundary ofthe set A.The metric projection of a point x ∈ H on the set A ⊂ H is givenby the formula PA(x) = a ∈ A : ‖x − a‖ = inf

y∈A‖x − y‖. We denote the

normal cone to a closed convex subset A ⊂ H at the point a ∈ A by N(A; a),i.e. N(A; a) = p ∈ H : (p, a) ≥ sup

x∈A(p, x).

For a subset A ⊂ H let UA() be the open -neighborhood UA() = x ∈H : inf

y∈A‖x− y‖ <

D e f i n i t i o n 1. A nonempty subset A ⊂ H is called strongly convex ofradius R > 0 if it can be represented as an intersection of closed balls of radiusR > 0, i.e. there exists a subset X ⊂ H such that A =

⋂x∈X

BR(x).

The analog of the next estimate was obtained in [1] for a subset in theHilbert space with C2 smooth boundary.

Theorem 1. Let A ⊂ H be a closed set. Let S ⊂ ∂A and Φ.=

( ⋃x∈S

N(A;x)

)⋂(H\UA(

Suppose that the function

δS(ε).= inf

δ > 0 : Bδ

(a0 + a12

)⊂ A, ∀a0, a1 ∈ S : ‖a0 − a1‖ = ε

satisfies δS(ε) ≥ Cεp, where C > 0, p ≥ 2.Then ∀ε > 0 ∃δ > 0 ∀x0, x1 ∈ Φ : ‖x0 − x1‖ < δ, [x0, x1] ⊂ Φ, ai =

PA(xi), i ∈ 0, 1, ‖x0 − a0‖ = ‖x1 − a1‖ ≥ , the following inequality holds:

‖x0−x1‖2 ≥ (1)

≥ ‖a0 − a1‖2 +(

8C

1− 21−p− ε

)‖a0 − a1‖p

+

(16C22

(1− 21−p)2− ε

)‖a0 − a1‖2p−2.

68

By Formula (1) for the case p = 2 we obtain that (under condition ofTheorem 1) for R = 1

8C

‖x0 − x1‖ ≥

√√√√((

R +

R

)2

− 2ε

)‖a0 − a1‖,

i.e. the estimate is precise, see [3]. In the case p > 2 the estimate is slightlybetter than the Lipschitz condition.

Theorem 2. Let A ⊂ H be a closed convex set. Let S ⊂ ∂A and Φ.=( ⋃

x∈SN(A;x)

)⋂(H\UA()). Suppose that ∃C ∈ (0; 1) such that ∀x0, x1 ∈ Φ :

[x0, x1] ⊂ Φ the following formula holds:

‖a0 − a1‖ ≤ C‖x0 − x1‖, ai = PA(xi), i ∈ 0, 1

Let ε ≤ C1−C and a ∈ S

Then if ∂A⋂Bε(a) ⊂ S then A

⋂Bε(a) is strongly convex set of radius

C1−C

The author was supported by grant RFBR 13-01-00295-a.

References

1. T.J. Abatzoglou. “The Lipschitz continuity of the metric projection” Journalof Approximation theory, 26, 212–218 (1979).

2. M.O. Golubev. “Metric projection in a Hilbert space and strong convexity”,Modern problems of function theory and their applications, Proceedings ofthe 16th Saratov Winter School. Saratov: LLC ”Publishing house” ScienceBook”, 2012. pp. 55–56.

3. M.V. Balashov, M.O. Golubev. “About the Lipschitz property of the matricprojection in the Hilbert space”, Journal of Mathematical Analysis and Ap-plications, 394, No. 2, 545–551 (2012).

69

An approach to solve the resource-constrainedproject scheduling problem

Evgeniy Goncharov1

1 Sobolev Institute of Mathematics, Novosibirsk, Russia; gon@math.nsc.ru

We consider the resource constrained project scheduling problem with prece-dence and resource constraints (RCPSP). We are given a set of activities J =1, ..., n. The partial order on the set of activities is given by directed acyclicgraph G = (V, E). Each activity j ∈ E is characterized by its processing timepj ∈ Z+ and the resource requirement for rjk(τ ) in resource type k on the timeinterval [τ −1, τ ), τ = 1, ..., pj . For each constrained resource type k are knownits capacity Rkt during the time interval [t − 1, t). All resources are renewable.Activities preemptions are not allowed. The objective is to compute the sched-ule S = sj that meets all resource and precedence constraints and minimizesthe makespan Cmax(S). A formal setting of the RCPSP problem is the following:

Cmax(S) = maxj∈E

(sj + pj) −→ minsj

(1)

Subject to:si + pi ≤ sj , i ∈ Pred(j), j ∈ E; (2)

∑

j∈U(t)

rjk(t− sj) ≤ Rkt , k ∈M, t = 1, ..., Tk; (3)

sj ∈ Z+, j ∈ E, (4)

where the set Pred(j) is the set of immediate predecessor for activity j ∈ U ,and U(t) = j | sj < t ≤ sj + pj is the set of the activities which are beingprocessed at the time instant [t− 1, t) at the schedule S.

Since the considered problem is NP-hard, it is expediently to design approx-imated algorithms for it with low time complexity.

We offer few heuristic algorithms in this paper. We made a deterministicgreedy algorithm up, which solves this problem with the time complexity de-pending on the number of activities as n log n [3]. This algorithm uses so-calledSerial schedule generation scheme, and its main idea is to use a solution of a re-laxed problem, where all constrained resources being accumulative. It is knownthat the problem with accumulative resources is polynomially solvable [1]. Forthe case of real processing times a polynomial time asymptotically optimal algo-rithm with an absolute error tending to zero for increasing number of activitieswas suggested by E.Kh.Gimadi [2]. We use this approximate algorithm to solveauxiliary relaxed problem.

70

We also offer a stochastic heuristic algorithm. It is designed as a series ofindependent trials randomized version of the greedy algorithm [3], where thesolution is taken as the best out of those obtained. This method is known asGreedy Randomized Adaptive Search Procedure (GRASP). In addition we rana procedure of local improvement on each solutions obtained (the so-called algo-rithm of Forward-backward improvement). Note that in this case the algorithmbecomes nonpolynomial.

Quality of these algorithms has been analyzed in a number of computationalexperiments using the standard sets j30, j60, and jl20 for the RCPSP from thePSPLIB-library. This paper provides a comparison of the stochastic heuristicalgorithm with the best algorithms of other authors.

We also offer approach to construct approximate hybrid algorithm for solv-ing the RCPSP based on greedy heuristics and approximate solution derivedfrom a accumulative resources problem.

The author is supported by the Russian Foundation for Basic Research(project no. 13-07-00809).

References

1. E.Kh.Gimadi, V.V.Zalyubovsky, S.V.Sevastianov Polynomial solvability ofthe project sheduling under stored resource constraints and given deadlines.// Discretnii analiz i issledovanie operatsii. Novosibirsk: Institut matematiki.Ser. 2, Vol. 7, N 1, 2000, 9–34 (in Russian)

2. E.Kh.Gimadi On some mathematical models and methods of planning oflarge-scale projects // Models and methods of optimization: Trudy AkademiiNauk SSSR, Siberian branch. Institut matematiki. Novosibirsk: Nauka, 1988.Vol. 10, P. 89–115 (in Russian)

3. E.N.Goncharov A greedy heuristic approach for the resource-constrainedproject scheduling problem // Studia Informatica Univeralis, 2012, Vol. 9,N. 3, pp. 79-90.

71

Numerical techniques for optimization problems with“expensive” functions

Alexander Gornov1, Anna Dorjieva1

1 Institute for System Dynamics and Control Theory of SB RAS, Irkutsk,Russia; gornov@icc.ru, ikadark@list.ru

Sometimes, in a number of real-world applications of extremal problemswe face quite complicated models, which require a tangible amount of com-putational resources for their numerical implementation. Such optimizationproblems arise when formalizing a series of models from nano- and geophysics,and certain processes in chemistry, biology, ecology and medicine. This class ofproblems also includes some “purely mathematical” problem statements, suchas, optimization of trajectory tubes of dynamical systems, optimal feedbackcontrol, worst-case design in dynamical systems with disturbances etc. Notethat a typical time (in relation to an average computational powers) for cal-culation of values of such expensive functions may vary between a number ofminutes and scores of hours [1].

According to a well-known classification, problems of this type are so-calledblack-box optimization problems [2], as it is usually impossible to calculate ana-lytically gradients of a cost function, even if the number of variables is smallish.In black-box optimization problems we usually deal with a lack of an a prioriinformation about properties of a cost function, and therefore, such problemsare naturally referred to as nonconvex problems. To our mind, the main diffi-culty in numerical analysis of optimization problems with “expensive” functionsconsists in a so-called functional redundancy of algorithms. Algorithms relia-bility, being the basical criterion in their software implementation, implies anexceed of a number of iterations, which leads to loss of algorithms’ efficiency.Numerical analysis of problems of this class is usually difficult, even by meansof best-of-known solvers.

In the present talk we propose some new approaches for design of numericalalgorithms for optimization with “expensive” cost functions. We claim thatalgorithms should fulfill the following criteria:

• They do not require more then a given (relatively small) number of eval-uations of the cost function;

• They should be suitable for parallel programming.

However, the real opportunity of applying optimization algorithms for a “hard”problem still implies their interactive software implementation as well as par-ticipation of a sufficiently qualified user.

72

The second purpose of the talk is to describe the work-in-progress solverOPTCON-T for optimization problems with “expensive” functions. This solveris supposed to encompass the software implementation of the mentioned algo-rithms. The current version of OPTCON-T includes the following modules:

• A database of algorithms for local and global optimization (which arecapable of working in both manual and automatic modes);

• An interface for low-level adjustment of algorithms’ parameters. Thisinterface is useful in design of multi-method computational schemes;

• A graphical kernel, which admits to employ special cognitive graphicsfacilities;

• A module for approximation of a cost function;

• Special computer aids for the database of calculated values of a cost func-tion.

Finally, we discuss some examples of optimization problems with “expen-sive” functions, and present test cases of both academical and practical values.

The authors were supported by the Russian Foundation for Basic Research(project no. 12-01-00193) and SB of RAS (interdisciplinary integration progectno. 83).

References

1. D.R. Jones, M. Schonlau, W.J. Welch . “Efficient Global Optimization ofExpensive Black-Box Functions,” Journal of Global Optimization, 1, No. 13,455–492 (1998).

2. T. Jansen “General Limits in Black-Box Optimization,” in: Analyzing Evo-lutionary Algorithms, Springer, Berlin, 2013, pp. 45–84.

73

Practical Approximation Algorithms for ReachableSet of Dynamic Controlled System

Alexander Gornov1, Evgeniya Finkelstein1

1 Institute for System Dynamics and Control Theory SB RAS, Irkutsk, Russia;gornov@icc.ru

The authors propose a number of methods to obtain external and internalestimates, as well as approximations of the boundary of the reachable set. Weconsider the method of stochastic approximation as the most reliable methodof giving an internal evaluation. Having received the first approximation bystochastic method in the form of a sufficiently compact and well-defined set,we can conclude on the possible use of algorithms for constructing the bound-aries of sets. The fastest of our developed algorithms for this purpose is theapproximation algorithm of the reachable set of a dynamic linear in controlsystem for small time intervals, which uses the property of bang-bang control.Another more universal method, but also better applicable for linear in con-trol systems is a method of maximizing the area bounded by a piecewise-linearcircuit, all of whose vertices are achievable points [1]. For non-linear dynamicsystems proposed search algorithm of the boundary points of the reachable set,based on the idea of the characterization of points obtained after integration,that is a boundary in case when it is impossible to solve the optimal controlproblem of transfer to a neighbourhood. Nevertheless, if the construction of theset boundary is not possible, can be applied technologies of external estimatesbased on simple geometric objects – ellipses, parallelepipeds, balls and others.We propose an algorithm for the reachable set approximating by ellipses union.

The conducted experiments confirmed the practicable computational effi-ciency of the proposed approaches and allowed us to estimate the scope of theimplemented algorithms.

The authors were supported by the Russian Foundation for Basic Research(project no. 12-01-00193)and Interdisciplinary Integration Project SB RAS no. 81.

References

1. Gornov A.Yu. Computational technologies for solving optimal control prob-lems, Nauka, Novosibirsk (2009).

74

General separation theorems of convex sets andconvex optimization problems

Valentin Gorokhovik

Institute of Mathematics, Minsk, Belarus; gorokh@im.bas-net.by

A subset H of a real vector space X is called a halfspace if both H and itscomplement X \H are convex. In the thirties of the last century S. Kakutaniand J.W. Tukey proved independently the following general separation theorem:any two convex sets A and B lying in X are disjoint if and only if there existsa halfspace H in X such that A ⊂ H and B ⊂ X \H. Later (in 1987) J.-E.Martinez-Legaz and I. Singer proved an analytical version of this theorem for thecase when X is a finite-dimensional vector space. However their approach cannot be extended directly to the infinite-dimensional case. In this talk we presnta new analytical version of the Kakutani—Tukey theorem that are suitableboth for finite-dimensional and infinite-dimensional cases and then we apply itto the study of some convex optimization problems. We begin with a classifica-tion of infinite-dimensional halfspaces by types. We show that in any infinite-dimensional vector space there exist halfspaces of three types: two (pointedand unpointed) Dedekind and non-Dedekind ones, while in finite-dimensionalspaces only two (pointed and unpointed Dedekind) types of halfspaces exist.Then we introduce and study a new class of real-valued functions called step-affine. Our main result is the following duality theorem: (i) a proper subsetC ⊂ X is a pointed (unpointed) Dedekind halfspace if and only if there existsa regular step-affine function u : X → R such that C = x ∈ X | u (x) ≥ 0(C = x ∈ X | u (x) > 0); (ii) a proper subset C ⊂ X is a non-Dedekindhalfspace if and only if there exists a singular step-affine function u : X → Rsuch that C = x ∈ X | u (x) > 0. Using this duality theorem we formulate ananalytical version of the Kakutani—Tukey theorem as a theorem on separationof convex sets by step-affine functions. As an application of this theorem we de-rive an analytical characterization of optimality for a convex vector optimizationproblem. Besides we prove a criterion of optimality for a convex programmingproblem which extends the well-known Kuhn—Tucker criterion to the problemsnot holding the Slater regularity condition.

75

Classes of positively homogeneous functions

Valentin Gorokhovik1, Marina A. Trafimovich2

1 Institute of Mathematics, National Academy of Sciences of Belarus, Minsk,220072, Belarus; gorokh@im.bas-net.by

2 Byelorussian State University, Minsk, 220050, Belarus; marvoitova@tut.by

In the talk we deal with positively homogeneous functions defined on afinite-dimensional vector space R

n. Recall, that a function p : Rn → R is said tobe positively homogeneous if p(λx) = λp(x) for all x ∈ R

n, λ > 0. The schemeof classification of positively homogeneous functions is presented on Fig.1.

The set of all positively homogeneous functions defined on Rn (P(Rn)) forms

the vector space with respect to the natural operations of pointwise additionand multiplication by a real number.

Five subspaces were extracted from the vector space of positively homo-geneous functions: the subspace of continuous positively homogeneous func-tions (PC(Rn)), the subspace of Lipshitzian positively homogeneous functions(PL(Rn)), the subspace of difference sublinear functions (PDC(Rn)), the sub-space of piecewise linear functions (PL(Rn)) and the subspace of linear functions(L(Rn)). These subspaces satisfy the following sequence of inequalities:

P(Rn) ⊃ PC(Rn) ⊃ PL(Rn) ⊃ PDC(Rn) ⊃ PL(Rn) ⊃ L(Rn). (1)

The following norms are introduced respectively on the first three subspaces:

‖p‖C = maxx∈S

|p(x)|, where S = x ∈ Rn| ‖x‖Rn = 1; (2)

‖p‖L = supx,y∈R

n

x 6=y

|p(x)− p(y)|‖x− y‖Rn

; (3)

‖p‖DC = inf‖p‖C + ‖p‖C |p = p− p, p, p : Rn → R — sublinear functions.(4)

The spaces (PC(Rn), ‖ · ‖C), (PL(Rn), ‖ · ‖L) and (PDC(Rn), ‖ · ‖DC) areBanach ones. Moreover, their norms satisfy the following inequalities:

‖p‖C ≤ ‖p‖L ∀p ∈ PL(Rn); (5)

‖p‖L ≤ ‖p‖DC ∀p ∈ PDC(Rn). (6)

76

Thus, there is an embedding of the space of Lipshitzian positively homoge-neous functions (PL(Rn), ‖ · ‖L) in the space of continuous positively homoge-neous functions (PC(Rn), ‖ · ‖C) and an embedding of the space of differencesublinear functions (PDC(Rn), ‖ · ‖DC) in the spaces of Lipshitzian positivelyhomogeneous functions (PL(Rn), ‖ ·‖L) and continuous positively homogeneousfunctions (PC(Rn), ‖ · ‖C).

Note that these three norms coincide on the space of linear functions L(Rn)and are equal to the standard norm on L(Rn).

positively homogeneous functions

continuous

positively homogeneous functions

Lipshitzian

positively homogeneous functions

difference sublinear functions

piecewise linear functions

linear functions

Fig. 1

The authors were supported by the National Program of Basic Researchersof Belarus.

77

Delimiting the Central Business District – aneconomic analysis using graph theory

Yana Grebenyuk1, Aleksandr Abramov2

1FEFU, Vladivostok, Russia; ya.grebenyuk@gmail.com2 FEFU, Vladivostok, Russia; alabramov2011@gmail.com

The city is object of studying of many scientific disciplines because allspheres of person’s activity are focused here: people live , work, realize theirabilities here. The city was a locus of concentration of history, culture, andbusiness since ancient times . Owing to continuous man activity, city, beingmultipurpose and flexible system, reacts to the changes happening in all spheresof public life. So some parts of urban environment, especially the market struc-ture, that have taken place over the recent decades, need new researches. Sothe Central Business District (CBD) as a concept of Urban Geography has beenexamined in various contexts. However, definitions of this mental construct arequalitative and no universal theory on its location and spatial extent in complexurban environments exists.

The definition was formulated on the theory Voishvillo, soD e f i n i t i o n CBD - space in the city, characterized by the concentration

of businesses and business services, developed infrastructure and high densityof buildings.

Let graph G = (V,E) – Vladivostok road system, where V the set of edges(roads), and E the set of vertices (cross-roads). Each edge has a weight c(e).The task of finding the location of the CBD Vladivostok can be formulated asthe problem of finding subgraph of G. The solution of the problem was foundat last step of the minimum weight cover algorithm.

References

1. E.K. Voishvillo. Ponyatie kak forma myshleniya: logiko gnoseologicheskiianaliz (Concept as a Form of Thinking: Logical and Epistemological Analy-sis), Izd. Mosk. Gos. Univ., Moscow (1989).

2. V.Kolmogorov . “Blossom V: A new implementation of a minimum cost per-fect matching algorithm,” Mathematical Programming Computation (MPC),1, No. 1, 43–67 (July 2009).

78

Ill-posedness and regularized algorithms for solvingquasi-variational inequalities

Milojica Jacimovic1, Nevena Mijajlovic1

1 University of Montenegro, Podgorica, Montenegro;milojica@jacimovic.me, nevenamijajlovic@hotmail.com

In our talk we will consider quasi-variational inequality:

find x∗ ∈ C(x∗)such that 〈F (x∗), z − x∗〉 ≥ 0 ∀z ∈ C(x∗),

where F : H → H is an operator of a Hilbert space H and C : H → 2H is apoint to set mapping with nonemoty closed and convex values.

It is known that, in general, quasi-variational inequalities are ill-posed. Thatis a small perturbations in initial data of a quasi-variational inequality couldproduce uncotrollable changes in the set of solutions of original problem (if thisset of solutions is nonempty). This fact generates the necessity to apply somemethods of regularization that will give satisfying approximation of solution.We will use the regularizations based on solving of quasi-variational inequality

find xα ∈ C(xα) such that 〈F (xα) + αR(xα, z − x∗〉 ≥ 0∀z ∈ C(xα).

where α > 0 and R : H → H is a regularizing operator such that F + αR hasbetter properties than F . More precisely, we study the convergence (as α→ 0)of the iterative processes of the gradient type combined with the regularizatioanapproach. Let us note that ;

References

1. A. S. Antipin, N. Mijajlovic, M. Jacimovic A Second-OrderIterative Method for Solving Quasi-Variational InequalitiesComputational Mathematics and Mathematical Physics, 2013, Vol. 53,No 3, pp. 258–264

2. Dj. Jovanov Existence of a solution of the quasi-varaitional inequalitywith semicontinuous operator, Yugoslav Journal of Opeartaions Research 16(2006), No 2, 147–152

3. A. A. Khan, C. Tammer Regularization of Quasi Variational Inequalities, Is-sues 2009-2010 of Report, Saale Institut fr Mathematik Halle, Inst. fr Math-ematik, 2010

79

Impulsive control of bifurcations under constraints

Vladimir Jacimovic1

1 Department of Mathematics, University of Montenegro, Podgorica,Montenegro;

Consider the dynamical system described by ODE’s, depending on thevector-parameter α ∈ Rs:

x = f(x, α), t ∈ [0, T ]. (1)

Assume that x = 0 is an equilibrium of the system for any α, i.e.

f(0, α) = 0, ∀α ∈ Rs.

Now, suppose that we are enabled to control the system (1) by small impulsivecontrols in the following way:

dx(t) = f(x(t), α)dt+G(t)dµ(t), t ∈ [0, T ]. (2)

Here, µ is k-dimensional Borel measure, such that µ(B) ⊆ K for all Borelsubsets B. The K is given convex, pointed cone in Rk. Therefore, the coneK represents the constraints of our control device (we can drive the systemonly in certain directions at the certain moments). The G : [0, T ] → Rn×k is acontinuous map.

In this report we will be concerned with the conditions of bifurcations inthe system (2). In another words, we will investigate conditions under whichthe system can be brought to bifurcation (at the critical value of parameterα = α0) by applying admissible controls µ. For instance, we will study ifthe system can be driven out of equilibrium into the periodical oscillations byapplying admissible impulsive controls µ. In all cases the bifurcation parameterwill be α.

This problem has numerous applications. We will also pay attention to morecomplicated dynamical systems (involving delays or reaction-diffusion systemsinstead of ODE’s) with bifurcations impulsive controls.

80

Massively parallel algorithms for two-side scaling oflarge sparse nonnegative matrices with applications

Igor Kaporin1, Olga Milyukova2

1 A.A.Dorodnitsyn Comput. Center R.A.S., Moscow, Russia;igorkaporin@mail.ru

2 M.V.Keldysh Inst. Appl. Math. R.A.S., Moscow, Russia;olgamilyukova@mail.ru

Given a nonnegative square n × n-matrix A, we consider the problem offinding two positive diagonal matricesX and Y such thatXAY is approximatelydoubly stochastic. This is equivalent to the numerical solution of nonlinearequation f(z) = 0, where f : R2n → R2n,

f(z) = ZBZe − e, B =

[0 AAT 0

], Z =

[X 00 Y

],

and e = [1...1]T ∈ R2n. The condition per(A) > 0 guarantees the existenceof approximate solution such that ‖f(z)‖1 < ω for any positive ω. Using tech-niques from [1], one can see that the latter condition holding with ω = 2 yieldsper(A) > 0. Therefore, approximate matrix scaling can be used, e.g., for theestimation of permanents as in [1] or for finding (suboptimal) solutions of thelinear assignment problem [2].

For large n the condition ‖f(z)‖1 < 2 implies rather fine scaling precision.This makes prohibitive the use of slowly converging simplest procedures suchas the Sinkhorn-Knopp algorithm [3]. In [4] and [5], Newton-type methodsfor the solution of the nonlinear equation f(z) = 0 are considered which mayconverge faster. Similarly, we propose to solve an equivalent logarithmic barrierminimization problem z = argminz>0 ϕ(z), where

ϕ(z) = xTAy − log det(XY ), X = Diag (x), Y = Diag (y).

Denoting gradϕ(z) = g(z), one can readily see that f(z) = Zg(z), which byz > 0 shows the required equivalence. The minimization is performed iterativelyusing the formula for the HessianH(z) = H(z)+Z−1Diag(g(z)) of the functionalϕ(z), where

H(z) =

[X−2 AAT Y −2

], H(z) =

[X−1Diag(Ay) A

AT Y −1Diag(ATx)

].

The latter matrix is symmetric nonnegative definite and has positive diago-nal. Moreover, the matrix ZT H(z)Z is diagonally dominant. Now we use the

81

Symmetric Gauss-Seidel preconditioning for H, that is,

H(z) = D + L+ LT , H(z) = (D + L)D−1(D + LT ).

Finally, we use the preconditioned steepst descent-type iterations

pk = H(zk)−1g(zk), zk+1 = zk − αkpk, αk = ϑgT (zk)pk/p

Tk H(zk)pk,

where ϑ = 0.85; the backtracking by αk is performed if necessary. NonlinearCG method was also tried though without any essential speedup.

The resulting algorithm has much faster (though linear) convergence as com-pared to those used, e.g., in [2]. Being formulated in terms of elementary vectorand matrix-vector product operations, it allows for an efficient massively par-allel implementation. Using the same test matrices as in [2], we present someresults obtained on MVS-100K multiprocessor.

The authors were supported by the Russian Foundation for Basic Research(project no. 11-01-00786), by the grant NSh-5264.2012.1, and by the Presidiumof Russian Academy of Sciences target program no. P-18.

References

1. N. Linial, A. Samorodnitsky, A. Vigderson. “Deterministic strongly polyno-mial algorithm for matrix scaling and approximate permanents,” Combina-torica, 20, 545–568 (2000).

2. M. Sharify, S. Gaubert, L. Grigori. “A parallel preprocessing for the optimalassignment problem based on diagonal scaling,” Preprint arXiv:1104.3830,22 pp. (2011). http://arxiv.org/pdf/1104.3830.pdf

3. R. Sinkhorn, P. Knopp. “Concerning nonnegative matrices and doublystochastic matrices,” Pacific J. Math., 21, 343–348 (1967).

4. M. Furer. “Quadratic convergence for scaling of matrices,” in:Proc. ALENEX/ANALC 2004 (L. Arge, G.F. Italiano, R. Segdewick,eds.) 2004, pp. 216-223.

5. P.A. Knight, D. Ruiz. “A fast algorithm for matrix balancing,” in: DagstuhlSeminar Proc. 07071, Web Information Retrieval and Linear Algebra Algo-rithms ( A. Frommer, M. W. Mahoney, D. B. Szyld, eds.) 18 pp. (2007).http://drops.dagstuhl.de/opus/volltexte/2007/1073

82

Multi-Memetic Global Optimization Based on theMind Evolutionary Computation

Anatoly Karpenko1, Mikhail Posypkin2, Anton Rubtsov3, MaximSakharov4

1 Bauman MSTU, Moscow, Russia; apkarpenko@mail.ru2 IPPI RAS, Moscow, Russia; mposypkin@gmail.com3 IPPI RAS, Moscow, Russia; rubtsov493@gmail.com

4 Bauman MSTU, Moscow, Russia; max.sfn90@gmail.com

This work presents a method of solving global optimization problems basedon the population Mind Evolutionary Computation (MEC) [1]. The algorithmcan be parallelized easily and can run on loosely-coupled computing systemswith distributed memory. This distinctive feature makes it promising to use ongrid-systems composed of personal computers (desktop grids) [2]. Since desktopgrids are widely spread nowadays due to relatively low cost the development ofalgorithms for this kind of systems is an important task.

MEC simulates some aspects of human behavior in the society. Each in-dividual in this algorithm is regarded as an intelligent agent, operating in thearbitrary group of people. In order to achieve a high score, an individual has tostudy under the most successful individuals of his group. The same principle isused in the intergroup competition.

Canonical algorithm was modified by authors to prevent the preliminaryconvergence. Operation of decomposition of the search region was added at thevery begining of the algorithm. Besides, hybridization of MEC with memeticalgorithm was proposed.

A memetic algorithm was introduced as a metaheuristic method that isbased on a certain combination of any population algorithm and one or morelocal optimization methods (memes). Overall scheme of the memetic algorithmis as follows. Group initialization S = (si, i ∈ [1 : |S|]), where si is the agent of apopulation and |s| is the total number of agents, takes place at the first iterationof the algorithm (t = 0). Then all operators of the population algorithm areapplied to every agent of the population S. On the basis of new positions ofevery agent si, local search is carried out, that is memes are launched fromthese positions. After the local search the quality of new positions of agentsis estimated. Then the iteration counter is incremented (t = t + 1) and thetermination conditions are tested.

Presented scheme of the memetic algorithm is flexible enough and allowsone to modify it in many different ways and, in particular, to create a multi-memetic algorithm. In this case the swarm of available memes is created M =

83

(mj , j ∈ [1 : |M |]); the most efficient of them then improves the current positionof the agent si. The key problems when developing multi-memetic algorithmsare selecting the most efficient meme and choosing best strategies to controlmemes application.

Authors present the C++ implementation of the MEC algorithm with thedescribed modifications. The application was developed so that it is possible touse it with various add-on libraries of memes.

Efficiency investigations of proposed multi-memetic algorithm and its im-plementation were carried out using five-dimensional benchmark functions. Theswarm of memes consisted of Solis-Wets method, Nelder-Mead algorithm, andquasi-Newton LBFGS method. Program implementations of these methodswere taken from BNB-Solver library [3]. Total computational cost and bestobtained mean value of the objective function were used as assessment criteriaof the algorithms efficiency.

The performance of the multi-memetic algorithm and its implementationwas also studied by the example of the problem of energy minimization fordifferent types of molecular clusters. Three types of molecular clusters werestudied; they are models of Lennard-Jones, Morse and Dzugutov. In each nu-merical experiment the global minimum was found to a high precision for everycluster composed of less than 25 particles.

Further development is aimed at the creation of the application that canrun on desktop grids.

References

1. A.P. Karpenko, A.E. Antukh. “Global optimization based on methods of par-ticle swarm, mind evolution and clonal selection”, technomag.edu.ru:Scienceand Education: Electronic Scientific and Technical Periodical, 12, No. 08,(2012).

2. D.P. Anderson. “Boinc: A system for public-resource computing and stor-age”, in: R. Buyya, editor, Fifth IEEE/ACM International Workshop onGrid Computing, 2004, pp. 4–10.

3. Y. Evtushenko, M. Posypkin, I. Sigal. “A framework for parallel large-scaleglobal optimization”, Computer Science - Research and Development, 23, No.03, 211–215 (2009).

84

A 2-approximation polynomial algorithm for asequence partitioning problem

Alexander Kel’manov1, Sergey Khamidullin2

1 Sobolev Institute of Mathematics, Novosibirsk, Russia; kelm@math.nsc.ru2 Sobolev Institute of Mathematics, Novosibirsk, Russia; kham@math.nsc.ru

We consider following strongly NP-hard [1]Problem 1-MSSCS-F. Given a sequence Y = (y1, . . . , yN ) of vectors from

Rq, and some positive integer numbers Tmin, Tmax and M > 1. Find a subset

M = n1, . . . , nM ⊆ N = 1, . . . , N such that

∑

j∈M‖yj − y(M)‖2 +

∑

i∈N\M‖yi‖2 → min,

where y(M) = 1|M|

∑i∈M yi, under constraints

1 ≤ Tmin ≤ nm − nm−1 ≤ Tmax ≤ N − 1, m = 2, . . . ,M,

on the elements of M.

In this work we present a 2-approximation efficient algorithm for the problem1-MSSCS-F. This algorithm implements a dynamic programming scheme andruns in time O(N2(M(Tmax − Tmin + 1) + q)), where M ≤ N and (Tmax −Tmin + 1) < N . Therefore, algorithm is polynomial and it finds the solution inO(N2(N2 + q)) time.

The authors were supported by the Russian Foundation for Basic Research(projects no. 12-01-00090, no. 13-07-00070, and no. 12-01-33028), and by theTarget Programme of the Siberian Branch of the Russian Academy of Sciences(integration projects no. 7B).

References

1. A.V. Kel’manov, A.V. Pyatkin. “On the complexity of some vector sequenceclustering problems”, Discrete Analysis and Operation Research, 20, No. 2,47–57 (2013). (in Russian).

85

A randomized algorithm for a clustering problem

Alexander Kel’manov1, Vladimir Khandeev2

1 Sobolev Institute of Mathematics, Novosibirsk, Russia; kelm@math.nsc.ru2 Novosibirsk State University, Novosibirsk, Russia;

vladimir.handeev@gmail.com

We consider the following clustering problem.

Problem 1-MSSC-F. Given a set Y = y1, . . . , yN of vectors from Rq

and a positive integer number M . Find a subset C ⊆ Y such that

∑

y∈C‖y − y(C)‖2 +

∑

y∈Y\C‖y‖2 → min,

where y(C) = 1|C|∑y∈C

y, under constrain |C| =M .

This problem is strongly NP-hard [1]. In [2], a 2-approximation algorithmfor the problem is presented; this algorithm runs in time O(qN2). In [3], apolynomial-time approximation scheme with a O(qN2/ε+1(9/ε)3/ε)-time com-plexity, where ε is an arbitrary relative error, is substantiated. In this workwe present a polinomial randomized algorithm which finds an εN -approximatesolution with probability 1− δN , where εN , δN → 0 as N → ∞.

The authors were supported by the Russian Foundation for Basic Research(projects no. 12-01-00090, no. 13-07-00070 and no. 12-01-33028).

References

1. A.V. Kel’manov, A.V. Pyatkin. “Complexity of Certain Problems of Search-ing for Subsets of Vectors and Cluster Analysis, ” Computational Mathemat-ics and Mathematical Physics, 49, No. 11, 1966–1971 (2009).

2. A.V. Dolgushev, A.V. Kel’manov. “An Approximation Algorithm for Solvinga Problem of Cluster Analysis, ” Journal of Applied and Industrial Mathe-matics, 5, No. 4, 551–558 (2011).

3. A.V. Dolgushev, A.V. Kel’manov, V.V. Shenmaier. “A PTAS for a problemof cluster analysis,” Intelligent Information Processing: Proc. of the 9thIntern. Conf., Montenegro, Budva, 242-244 (2012). (In Russian).

86

On a sequences summation problem

Alexander Kel’manov1, Liudmila Mikhailova2

1 Sobolev Institute of Mathematics, Novosibirsk, Russia; kelm@math.nsc.ru2 Sobolev Institute of Mathematics, Novosibirsk, Russia; mikh@math.nsc.ru

We consider the following optimizationProblem SS (Sequences Summation). Given numerical sequences g(n)

and f(n), where n ∈ N = 1, . . . , N, and some positive integer numbers J, K,Tmin, and Tmax. Find two disjoint tuples K ⊆ N and J ⊆ N containing theelements numbers from these sequences such that |K| = K, |J | = J, and

G(K,J ) =∑

k∈Kg(k) +

∑

j∈Jf(j) → max,

under constraints on the elements of united tuple K∪J = n1, . . . , nK+J, i.e.

Tmin ≤ nm − nm−1 ≤ Tmax ≤ N − 1, m = 2, . . . ,K + J.

The exact efficient algorithm for the SS problem is proved in [1]. This algo-rithm runs in O(N5) time. Here we present a new exact polynomial algorithm.It implements a dynamic programming scheme and builds a solution in O(N4)time. So, the presented algorithm is N times faster then already existing one.

The authors were supported by the Russian Foundation for Basic Research(projects no. 11-01-00696, no. 12-01-00090, no. 13-07-00070, and no. 12-01-33028), and by the Target Programme of the Siberian Branch of the RussianAcademy of Sciences (integration projects no. 7B and no. 21A).

References

1. A.V. Dolgushev, A.V. Kel’manov. “An algorithm for noiseproof recognitionof a sequence that includes a given repeated vector and interfering vector-insertions from the given alphabet”, Proc. of the 14-th Russian Conf. ”Math-ematical Methods of Pattern Recognition”, 225–228 (2009). (in Russian).

87

A fully polynomial time approximation scheme for aproblem of choosing a vector subset

Alexander Kel’manov1, Semyon Romanchenko2

1 Sobolev Institute of Mathematics, Novosibirsk, Russia; kelm@math.nsc.ru2 Sobolev Institute of Mathematics, Novosibirsk, Russia; rsm@math.nsc.ru

We consider following strongly NP-hard [1]Problem VS-2. Given a set Y = y1, . . . , yN of vectors from R

q, andinteger M > 1. Find a subset C ⊆ Y such that

∑

y∈C‖y − y(C)‖2 → min,

where y(C) = 1M

∑y∈C y, under constrain |C| =M .

The following results for VS-2 problem are known: a 2-approximation al-gorithm having a O(qN2)-time complexity [2]; an exact pseudopolynomial al-gorithm having a O(qN(2MB)q)-time complexity (for the case when vectorcomponents have integer values and the dimension of space is fixed), where Bis the maximum absolute value of input vectors coordinates [3]; a PTAS havinga O(qN2/ε+1(9/ε)3/ε)-time complexity, where ε is a relative error [4]. In thiswork we present a FPTAS for the case of the fixed dimension of space. Ouralgorithm builds an (1+ ε)–approximate solution in O(N2(M/ε)q) time for anyε > 0.

References

1. A.V. Kel’manov, A.V. Pyatkin. “NP-Completeness of Some Problems ofChoosing a Vector Subset,” Journal of Applied and Industrial Mathematics,5, No. 3, 352–357 (2011).

2. A.V. Kel’manov, S.M. Romanchenko. “An Approximation Algorithm forSolving a Problem of Search for a Vector Subset,” Journal of Applied andIndustrial Mathematics, 6, No. 1, 90–96 (2012).

3. A.V. Kel’manov, S.M. Romanchenko. “Pseudopolynomial Algorithms forCertain Computationally Hard Vector Subset and Cluster Analysis Prob-lems,” Automation and Remote Control, 73, No. 2, 349–354 (2012).

4. V.V. Shenmaier “Approximation scheme for one problem of a vector subsetchoice”, Diskretn. Anal. Issled. Oper., 19, No. 2, 92–100 (2012) (in Russian).

88

On complexity of one maximum cut problem

Alexander Kel’manov1, Artem Pyatkin2

1 Sobolev Institute of Mathematics, Novosibirsk, Russia; kelm@math.nsc.ru2 Sobolev Institute of Mathematics, Novosibirsk, Russia; artem@math.nsc.ru

The following problem is considerd:

Problem Max-Cut(Rq)-SD (Max-Cut in Euclidean space, the case forSquared Distances). Input : A set X = x1, . . . , xN of Euclidean vectors R

q

and positive number A. Question: Is there a partition of the set X into twosubsets Y and Z such that

∑y∈Y

∑z∈Z ‖y − z‖2 ≥ A?

The optimization variant of this problem corresponds to the search for a cutof maximum weight in the complete undirected weighted graph whose verticesare Euclidean points and the weights of the edges are equal to the squareddistances between these points. It is known that in the general (non-Euclidean)case when the weights of the edges in the graph are positive the problem isstrongly NP-complete [1]. The complexity status of the considered (Euclidean)case was not earlier stated. This was the motivation of our research. Besides,this problem is actual, in particular, for solving noise proof clustering problemsin data analysis. The main result of the work is:

Theorem 1. Problem Max-Cut(Rq)-SD is strongly NP-complete.

It is shown in the proof that the classical strongly NP-complete problemMax-Cut can be reduced to the partial case of the considered problem with thenumerical parameters bounded by a polynomial on the input size.

The work was supported by the RFBR grants no. 12-01-00090, no. 12-01-00093, no. 12-01-00184, no. 12-01-33028, no. 13-07-00070, and by the grants ofthe aim SB RAS program (integration projects 7B and 21A).

References

1. Garey M.R., Johnson D.S. Computers and Intractability: A Guide to theTheory of NP-Completeness. San Francisco: Freeman, 1979. 314 P.

89

Algorithms for constructing the Lattices of Cubes andtheir applicability for solving optimization problems

Ruben.V. Khachaturov1

1 Dorodnicyn Computing Centre of RAS, Moscow, Russia; rv khach@yahoo.ie

Algorithms for constructing the Lattices of Cubes (Cubes Lattices) [1, 2] aredescribed, the results of application of these algorithms for various dimensionsof lattices are illustrated (fig. 1). The total number of elements of such lattice(including the empty set) is equal to 3M + 1, where M is dimension of thelattice. It is proved, that the Lattice of Cubes is a lattice with supplement,that allows to use effective algorithms [3, 4, 5] for solving on it the problems ofminimization and maximization of supermodular functions. Concrete examplesof such functions are given. The possibilities of setting and solving a new classof problems on the Cubes Lattices are discussed.

Theorem.The Cubes Lattice K is a lattice with supplement.Proof. To prove this theorem it is enough to show, that for any element

of the lattice C1 ∈ K there will be at least one element C2 ∈ K such thatC1 ∪ C2 = (0; I), C1 ∩ C2 = ∅.

Indeed, let C1 = (ω11;ω

12), where ω

11 ⊂ ω1

2 , and C2 = (ω21 ;ω

22), where ω

21 ⊂

ω22 . Let’s show, that the element of the lattice C2, at which ω2

1 = 0 and ω22 =

I \ ω12 , is a supplement to the element C1.It can be proved by direct check, using the above-stated definitions of union

and intersection operations for elements of the Cubes Lattice K, as follows:

C1 ∪ C2 = (ω11 ;ω

12) ∪ (0; I\ω1

2) = (0; I).

C1 ∩ C2 = (ω11;ω

12) ∩ (0; I\ω1

2) = (ω11 ; 0) = ∅.

The theorem is proved.

Two algorithms for constructing diagrams of a set of all C ∈ K (i.e. —Lattice of Cubes) have been worked out [1, 2].

Figure 1 shows the constructed Lattice of Cubes (Cubes Lattice) for M = 4.

90

Fig. 1. The Cubes Lattice diagram for M = 4

References

1. V.R. Khachaturov, R.V. Khachaturov. “Lattice of Cubes,” Journal of Com-puter and Systems Sciences International (Izvestiya RAN), 47, No. 1, 40–46(2008).

2. V.R. Khachaturov, R.V. Khachaturov. “Cubes Lattices and supermodularoptimization,” in: Works of the third international conference devoted tothe 85-anniversary of the correspondent member of the Russian Academy ofSciences, professor L.D. Kudryavtsev, publishing house of MFTI, Moscow,pp. 248–257 (2008).

3. V.R. Khachaturov. Mathematical Methods of Regional Programming,Economic-mathematical library, Nauka, Moscow (1989).

4. V.R. Khachaturov, V.E. Veselovskiy, A.V. Zlotov, et al. Combinatorial Meth-ods and Algorithms for Solving the Discrete Optimization Problems of theBig Dimension, Nauka, Moscow (2000).

5. V.R. Khachaturov, R.V. Khachaturov, R.V. Khachaturov. Supermodular Pro-gramming on Finite Lattices. Journal of Computational Mathematics andMathematical Physics, Vol. 52, No. 6, pp. 855-878 (2012).

91

Boosting approximation algorithm for minimumaffine separating committee problem

Michael Khachay1

1 Krasovsky Institute of Mathematics and Mechanics, Ekaterinburg, Russia;mkhachay@imm.uran.ru

An affine separating committee is a natural discrete generalization of sep-arating hyperplane onto the case of inseparable finite sets. Let A,B ⊂ Rn,A ∪ B = m and f1, . . . , fq : Rn → R be affine functions. The finite sequenceK = (f1, . . . , fq) is called an affine separating committee for A and B if

|i ∈ Nq : fi(a) > 0| >q

2(a ∈ A),

|i ∈ Nq : fi(b) < 0| >q

2(b ∈ B).

The number q is called a length of the committee K, and the sets A and B arecalled separable by K.

The MASC P r o b l e m. For given A,B ⊂ Qn it is required to find anaffine separating committee K of minimum possible length.

Theorem 1 [1]. MASC is NP-hard and remains intractable even if A∪B ⊂x ∈ 0, 1, 2n : ‖x‖2 6 2 . MASC belongs to Apx approximability class.

We consider the MASC-GP(n) subclass of MASC problem, where the setA ∪B ⊂ Qn being in the general position and dimensionality n is fixed.

Theorem 2 [2]. MASC-GP(n) is solvable in polynomial time, if n = 1 andNP-hard, otherwise.

To the moment, the most accurate polynomial time approximation algo-rithm for MASC-GP(n) was GreedyCommittee algorithm [2] with approxima-tion guarantee of O(m/n). Using the boosting strategy, we propose a new poly-nomial time approximation algorithm BGC with guarantee of O((m/n lnm)1/2.

Let xε be minimax first-player strategy in positional zero-sum game con-sidered in [3]. We use it as a subroutine. Define a partial function Boost :(0, 1/2) × (0, 1) → true, false × Y, where Y = Y1 × Y2 × . . . is a feasiblestrategies set of the second player, Boost(γ, ε) = (S(γ, ε), R(γ, ε)), and func-tions S and R are calculated (simultaneously) while playing the game, wherethe first player chooses the strategy xε, and the second — any y ∈ Y. Whilethe function S(γ, ε) is defined for arbitrary γ ∈ (0, 1/2) and ε ∈ (0, 1) andtakes the value true iff the game was played successfully and loss of the firstplayer L(x, y) ≤ ε, the function R(γ, ε) is defined (and equal to y) if and onlyif S(γ, ε) = true.

92

Suppose t = ⌈ ⌊(m−n)/2⌋n

⌉, ε = 1m+1

, and lnm < 2t+ 1.A l g o r i t h m BGC.

1. Find

t0 = argmin

t ∈ 1, . . . , t : S

(1

4t+ 2, ε

)∧((t = 1) ∨ S

(1

4t− 2, ε

))

2. If t0 6 1/2(((2t+ 1)/(2 ln((m+ 1)/2)))1/2 − 1

), define γ = 1/2(2t0 + 1)−1,

y = R(γ, ε), y = (M1, . . . ,Mk), and Mi = A′i ∪ B′

i, where A′i ⊂ A, B′

i ⊂ B.Output the sequence K = (f1, . . . , fk), where fi is an affine function separatingA′i and B

′i.

3. Otherwise, output the result of GreedyCommittee.

Theorem 3. BGC is a correct algorithm and find an approximate solutionof MASC-GP(n) problem in O(mn+3/n lnm) + ΘGC with accuracy

BGC(A,B)

OPT(A,B)6 ⌈2m ln((m+ 1)/2)⌉1/2, where m = 2t + 1.

Here ΘGC is time complexity of GreedyCommittee.The author was supported by the Russian Foundation for Basic Research

(projects no. 13-01-00210 and 13-07-00181) and Ural Branch of RAS (projectsno. 12-P1-1016 and 12-S1-1017/1).

References

1. M. Khachai. “Computational and approximational complexity of combina-torial problems related to committee polyhedral separability of finite sets”,Pattern Recognition and Image Analysis, 18, No. 2, 237–242 (2008).

2. M. Khachai, M. Poberii. “Complexity and approximability of committee poly-hedral separabiility of sets in general position”, Informatica, 20, No. 2, 217–234 (2009).

3. Y. Freund “Boosting a weak algorithm by majority ”, Information and Com-putation. 121, 256–285 (1995).

93

Optimal control of boundary value problems on bothends of the time interval

Elena Khoroshilova

Lomonosov Moscow State University, Moscow, Russia;khorelena@gmail.com

There is considered an optimal control problem for a fixed time interval[t0, t1] with a free right end. The dynamics of absolutely continuous trajectoriesx(·) is described by a system of linear differential equations. Controls u(·) ∈ Uare assumed to be bounded in the norm of Lr2[t0, t1].

The problem is to determine such a control u∗(·) that the correspondingtrajectory x∗(·) has being started from point x(t0) = x∗

0 and reached the pointx(t1) = x∗

1. The important thing is that the initial and terminal values oftrajectory are given implicitly as the solutions of convex optimization problemswith linear boundary constraints and have to be found:

x∗0 ∈ Argminϕ0(x0) | A0x0 ≤ a0, x0 ∈ Rn,

x∗1 ∈ Argminϕ1(x1) | A1x1 ≤ a1, x1 ∈ Rn,

x(t) = D(t)x(t) +B(t)u(t), x(t0) = x∗0, x(t1) = x∗

1 ,x(·) ∈ ACn[t0, t1], u(·) ∈ U,

where ϕ0(x0), ϕ1(x1) are convex differentiable functions with gradients satisfy-ing the Lipschitz condition.

To solve this problem we have proposed a method using the Lagrangian

L(p0, p1, ψ(·);x0, x1, x(·), u(·)) = 〈∇ϕ0(x∗0), x0〉+ 〈∇ϕ1(x

∗1), x1〉

+〈p0, A0x0 − a0〉+ 〈p1, A1x1 − a1〉+∫ t1

t0

〈ψ(t), D(t)x(t)+B(t)u(t)− d

dtx(t)〉dt,

for all (p0, p1, ψ(·)) ∈ Rm+ × Rm+ × Ψn2 [t0, t1], (x0, x1, x(·), u(·)) ∈ Rn × Rn ×ACn[t0, t1]× U.

According to the analog of Kuhn-Tucker theorem, if (x∗(·), u∗(·)) is therequired solution, then there exist dual variables (p∗0, p

∗1, ψ

∗(·)) such that (p∗0, p∗1,

ψ∗(·);x∗(·), u∗(·)) is a saddle point of the Lagrange function. This allows us toreduce the original problem to finding a saddle point of the Lagrangian [1].

To solve the resulting boundary value problem we used the method of ex-tragradient type. Each iteration of the method is divided into two half-step:

94

1) projected half-step

d

dtxk(t) = D(t)xk(t) +B(t)uk(t), xk(t0) = xk0 ,

pk1 = π+(pk1 + α(A1x

k1 − a1)),

d

dtψk(t) +DT(t)ψk(t) = 0, ψk1 = ϕ1 + AT

1 pk1 ,

uk(t) = πU (uk(t)− αBT(t)ψk(t)),

pk0 = π+(pk0 + α(A0x

k0 − a0)),

xk0 = xk0 − α(ϕ0 + AT0 p

k0 + ψk0 );

2) basic half-step

d

dtxk(t) = D(t)xk(t) +B(t)uk(t), xk(t0) = xk0 ,

pk+11 = π+(p

k1 + α(A1x

k1 − a1)),

d

dtψk(t) +DT(t)ψk(t) = 0, ψk1 = ϕ1 + AT

1 pk1 ,

uk+1(t) = πU (uk(t)− αBT(t)ψk(t)),

pk+10 = π+(p

k0 + α(A0x

k0 − a0)),

xk+10 = xk0 − α(ϕ0 + AT

0 pk0 + ψk0 ), k = 0, 1, 2...

The convergence of the method was proved: the weak convergence in con-trols and the strong convergence in trajectories and dual trajectories.

The author was supported by the Russian Foundation for Basic Research(project no. 12-01-00783) and by the Program ”Leading Scientific Schools”(NSh-5264.2012.1).

References

1. E.V. Khoroshilova. “Extragradient method of optimal control with terminalconstraints”, Automation and Remote Control, 73, No. 3, 517–531 (2012).

95

Generalized Chaplygin problem

Yuri Kiselev1, Sergey Avvakumov2

1 Lomonosov Moscow State University, Moscow, Russia; kiselev@cs.msu.su2 Lomonosov Moscow State University, Moscow, Russia; asn@cs.msu.su

The following two-dimensional optimal control problem is studied:

x = u; x ∈ IR2, u ∈ U ⊂ IR2; 0 ≤ t ≤ T ;

x(0) = x(T ) = a ∈ IR2, x(0)/‖x(0)‖ = q(α0);

L[u] ≡∫ T

0

(A∗x, u

)dt −→ max

u(·).

(1)

Here q(α) = ( cosαsinα ) is a unit vector, A =

(0 1−1 0

)is the (2× 2)-matrix, A∗ is the

transposed matrix A. The collection of known parameters T , U , a, α0 ∈ [0, 2π)is given. The control domain U is a plane smooth convex compact set, 0 ∈ intU ;support function c(ψ) = max

u∈U(u, ψ) of the set U and its distant functions play

the important technical role for problem (1) solution. It is assumed that c(ψ) >0 ∀ψ 6= 0, gradient c′(ψ) and Hessian c′′(ψ) are defined and continuous forall ψ 6= 0, rank c′′(ψ) = 1 for ψ 6= 0, [2]. Geometric sense of the cost L isdoubled area of the plane figure, which is limited by the closed curve x = x(t),0 ≤ t ≤ T . In the original Chaplygin’s problem [1] about maximal flyby area(single anticlockwise bypass) the control domain U is a circle with translatedcenter, 0 ∈ intU .

The Pontryagin maximum principle is used for solving the problem (1). Inthe study of the maximum principle boundary value problem

x = c′(A∗x+ ψ), x

∣∣t=0

= x∣∣t=T

= a, x(0)/‖x(0)‖ = q(α0),

ψ = A∗c′(A∗x+ ψ),

the theorem about support function’s gradient and the vector first integral A∗x−ψ = const are used. The important role belongs to the following two-dimensionalHamiltonian system

p = A∗c′(p), p∣∣t=0

= p0, p, p0 ∈ IR2 \ 0

analysis. Solution of this system admits certain description in an analyticalform.

In the formulation of final result the polar set U of the convex compact set Uis involved. Optimal motion is running along the curve obtained from the polar

96

curve ∂U as result of some linear transformations depending on the parametersof the problem (1).

Support function’s technique is useful in construction of numerical algo-rithms for optimal solution search and also in theoretical investigations [3-10].In conclusion it is possible to draw attention to short publication [4] concerningChaplygin’s problem.

The authors were supported by the RFBR (no. 12-01-00175) and PSLSS(no. 65590.2010.1).

References

1. M.A. Lavrent’ev, L.A. Lyusternik. A course in variational calculus, ONTI,Moscow-Leningrad (1938) (In Russian).

2. Yu.N. Kiselev, S.N. Avvakumov, M.V. Orlov. Optimal control. Linear theoryand applications, MAKS Press, Moscow (2007) (In Russian).

3. Yu.N. Kiselev. “Generalized Chaplygin’s problem,” Proceedings of the TenthCrimean Autumn Mathematical School, 10, 160–163 (2000).

4. Yu.N. Kiselev. “Optimal design in a smooth linear time-optimal problem,”Differential Equations, 26, No. 2, 174–178 (1990).

5. S.N. Avvakumov, Yu.N. Kiselev, M.V. Orlov. “Methods for solving optimalcontrol problems, based on Pontryagin’s maximum principle,” Trudy Mat.Inst. Steklov, Optim. Upr. i Differ. Uravn., 211, 174–178 (1995).

6. Yu.N. Kiselev. “Rapidly converging algorithms for solving time-optimalityproblem,” Kibernetika (Kiev), No. 6, 47–57, 62 (1990) (In Russian).

7. Yu.N. Kiselev, M.V. Orlov. “Numerical algorithms for linear time-optimalities,” Zh. Vychisl. Mat. i Mat. Phys., 31, No. 12, 39–43 (1991)(In Russian).

8. S.N. Avvakumov, Yu.N. Kiselev. “Support functions of some special sets,constructive smoothing procedures, and geometric differences,” Problems indynamic control; Mosk. Gos. Univ. im Lomonosova, Fak. VMK, Moscow,No. 1, 24–110 (2005) (In Russian).

9. S.N. Avvakumov, Yu.N. Kiselev. “Some algorithms of optimal control,” Proc.Steklov Ins. Math, Control; Stability, and Inverse Problems of Dynamics,suppl. 2, Ekaterinburg, 12, No. 2, 3–17 (2006) (In Russian).

10. Yu.N. Kiselev. “Construction of exact solutions for the nonlinear time-optimality problem of a special type,” Fundam. Prikl. Math., 3, No. 3,847–868 (1997) (In Russian).

97

On solving some production groups formationproblems based on discrete optimization

Alexander Kolokolov1, Lubov Afanasyeva2

1 Omsk Branch of Sobolev Institute of Mathematics SB RAS, Omsk, Russia;kolo@ofim.oscsbras.ru

2 F.M. Dostoevsky State Omsk University, Omsk, Russia;lubovshulepova@gmail.com

The production groups formation problems are complex and actual so it isnecessary to use models and methods of discrete optimization to solve them. Anumber of statements of such problem based on well-known assignment problemare presented in papers [1]. The paper [2] includes issues of design productionteams related to sets coverings and plant locations.

In this paper we research production groups formation problem with maxi-mized level of comfort between specialists and without strained relations. Notethat this problem has been studied in [3]. Graph-theoretic statements and lin-ear integer programming models are proposed and analysed, NP -hardness ofthis problem is proved as well as its polynomial reduction to the maximal inde-pendent set problem. To solve problem algorithms are proposed and realized,computational experiment is carried out. It showed prospectivity of applyingthe developed models and algorithms in decisions making support systems.

The authors were supported by the Russian Foundation for Basic Research(project no. 13-01-00862).

References

1. L.D. Afanasyeva, A.A. Kolokolov “Design and analysis of algorithm for solv-ing some formation of production groups problems”, Omsk Scientific Bulletin,110, No. 2, 39–42 (2012).

2. A.A. Kolokolov, A.V. Artemova, L.D. Afanasyeva “Solution of some humanresource problems using optimization methods”, in: Materials of VIII Intern.scientific and engineering. conf. “Dynamics of systems, tools and machines”,Publishing House of Omsk State Technical University, Omsk, 2012, pp. 55–58.

3. A.A. Kolokolov, L.D. Afanasyeva “Research of Production Groups Forma-tion Problem Subject to Logical Restrictions”, Journal of Siberian FederalUniversity, 6, No. 2, 145–149 (2013).

98

The control of equilibrium at highly monopolizedmarkets

Ekaterina Konovalova1, Anatoly Panyukov2

1 South Ural State University, Chelyabinsk, Russia;konovalova ekaterina@bk.ru

2 South Ural State University, Chelyabinsk, Russia; a panyukov@mail.ru

Let us consider monopolized markets as bimatrix game:

Γ = 〈M,Π, XM,XΠ, HM,HΠ〉, (1)

with

• monopolist M;

• setXM of monopolists strategies, namely monopolists supply to customersusing technology i = 1, 2, · · · , n (determine the product and its price Pi);

• set Π of customers;

• set XΠ of customers strategies, namely customers demand for productwith its quality (determine consumption Vj);

• monopolist’s payoff matrix HM = PV T − ∆M with elements (HM)ij =PiVj − (∆M)ij monopolist’s profits in case (i, j) where Pi is monopolyprice, Vj is consumption at monopoly price matrix ∆M of governmentinfluence for the monopolist;

• customers payoff matrix HΠ = −PV T − ∆Π with elements HΠij =−PiVj + (∆Π)ij that is total utility for consumers given case, matrix∆Π of government influence for the customers.

Let on the stage K of the monopolist and customers game with given gov-ernment influence (i.e. matrix ∆

(k)M and ∆

(k)Π ) there is a market equilibrium

situation. This situation is the result of the evolution (i.e. vectors P and Vtake values P (k) and V (k) respectively). It may no longer satisfy the needs of thegovernment for economic performance or other reasons. Government sets thenew right equilibrium (i∗, j∗) by changing the matrix ∆

(k)M and ∆

(k)Π to matrix

∆M and ∆Π.The formation of new payoff matrix HM = P (k)V (k)T − ∆M and HΠ =

−P (k)V (k)T +∆Π (such that the situation (i∗, j∗) is Stackelberg equilibrium inpure strategies) reduced to formation such matrix ∆M and ∆Π as:

Σi∈I,j∈J (uΠ(i, j) + uM (i, j)) → minu,∆

, (2)

99

(∀i ∈ I, j ∈ J)(P (i)V (j)−∆M (i, j) ≤ P (i∗)V (j)−∆M (i∗, j)), (3)

(∀i ∈ I, j ∈ J)(−P (i)V (j) +∆Π(i, j) ≤ −P (i)V (j∗)−∆Π(i, j∗)), (4)

Σi∈I,j∈J∆M (i, j) = Σi∈I,j∈J∆Π(i, j), uM , uΠ ≥ 0, (5)

(∀i ∈ I, j ∈ J)(−uΠ(i, j) ≤ ∆Π(i, j) ≤ uΠ(i, j)), (6)

(∀i ∈ I, j ∈ J)(−uM (i, j) ≤ ∆M (i, j) ≤ uM (i, j)). (7)

P r o p o s i t i o n 1. The problem (2) - (7) has an optimal solution

References

1. E.D. Konovalova. “Performance analysis of government control methodsadaptability at temporal high-rate monopolization market state changes,” in:Mathematical and statistical study of the socio-economic processes, SouthUral State University, Chelyabinsk, 2011, pp. 5–12.

2. E.D. Konovalova. “The development of statistical tools for the expert sur-vey,” in: Statistics. Modeling. Optimization, South Ural State University,Chelyabinsk, 2011, pp. 311–315.

3. A.V. Panyukov, E.D. Konovalova. “Performance analysis of government con-trol adaptability at temporal high-rate monopolization market state changes,”in: Perm University Herald special issue Economy , Perm University, Perm,2012, pp. 58–68.

100

Selection of the optimal operation mode of thespillway of Boguchanskaya HPP in winter 2012-2013

Vladimir N. Koterov1

1 Institution of Russian Academy of Sciences Dorodnicyn Computing Centre ofRAS, Moscow, Russia; vkoterov@yandex.ru

Work of spillways of large hydroelectric power stations is usually accompa-nied by formation of a large number of air-water mixtures (condensed moisture).Under adverse conditions, the movement of this water-air cloud at working sitesof hydroelectric power station can complicate the completion of the constructionworks and operation of power equipment.

The report presents the results of research aiming at the selection of theoptimal working mode of the spillway of Boguchanskaya hydroelectric stationon the Angara River in winter 2012-2013, in which the negative effect of thementioned above phenomenon would be minimal. This work is based on a three-dimensional computer model of forced convection air and condensed moisturetransfer, developed in the 2005-2007 [1-4]. In particular, the model was appliedto the scientific justification of measures to ensure the safe operation of theSayano-Shushenskaya hydroelectric station spillway in emergency period 2009-2010.

References

1. V. N. Koterov, B. V. Arkhipov. V. V. Belikov, V. V. Solbakov, V.E. Fedosov.“Numerical modeling of the formation and transport of condensed moisturein the lower pool of a hydroproject during the operation of spring-boardspillways,” Power Technology and Engineering, 41, No.5, 255–264 (2007).

2. V. N. Koterov, B. V. Arkhipov. V. V. Belikov, V. V. Solbakov. “Droplet watertransport in tail water of hydroelectric station at working of jump spillway,”Journal of Mathematical Modeling, 20, No.5, 78–82 (2008).

3. V. N. Koterov, V. V. Belikov. “Investigation and modeling of thermal airconvection and transport of local precipitation during winter operation of theservice spillway at the Sayano-Shushenskaya HPP,” Power Technology andEngineering, 46, No. 3, 190–197 (2012).

4. V.N. Koterov. Education and droplet water transport at working of HPP spill-way (theory and practice of mathematical modeling of natural and industrialprocesses), Lambert Academic Publishing, Saarbrucken (2012).

101

Identifying suspicious efficient units in DEA models

Vladimir Krivonozhko1, Finn Førsund2, Andrey Lychev3

1 National University of Science and Technology “MISiS”, Moscow, Russia;KrivonozhkoVE@mail.ru

2 University of Oslo, Oslo, Norway; F.R.Forsund@econ.uio.no3 National University of Science and Technology “MISiS”, Moscow, Russia;

Lytchev@mail.ru

After a decade of applications of Data Envelopment Analysis (DEA), orig-inating in [1] and generalized and put into the linear programming format weuse today by Charnes et al. [2], it was recognized that results both concerningefficiency scores and shape of the frontier production function, on which Farrellefficiency measures are based, were not always adequate when confronted withexpert knowledge of the units to which DEA was applied. Types of inadequaciesdiscussed in the literature have been that too many efficient units may appearin some DEA models, a DEA model may show an inefficient unit from the pointof view of experts as an efficient one.

The purpose of this paper is to identify units that may unduly becomeefficient. The concept of a terminal unit is introduced for such units. It isshown by establishing theorems how units can be identified as terminal units.An approach for improving the adequacy of DEA models based on terminalunits is suggested. We consider the BCC model [3] as a basic model in ourpaper, since BCC model can approximate [4] any DEA model from a largefamily of DEA models.

References

1. M.J. Farrell. “The measurement of productive efficiency,” Journal of theRoyal Statistical Society, Series A (General) 120, 253–281 (1957).

2. A. Charnes, W.W. Cooper, E. Rhodes. “Measuring the efficiency of decisionmaking units,” European Journal of Operational Research, 2, 429–444 (1978).

3. R.D. Banker, A. Charnes, W.W. Cooper “Some models for estimating tech-nical and scale efficiency in data envelopment analysis,” Management Science,30, 1078-1092 (1984).

4. V.E. Krivonozhko, O.B. Utkin, M.M. Safin, A.V. Lychev. “On some gener-alization of the DEA models,” Journal of the Operational Research Society,60, 1518-1527 (2009).

102

Computing modeling and investigation of a surfacebarrier discharge

Vladimir Krivtsov1, Victor Soloviev2

1 CCAS, Moscow, Russia; krivtsov@ccas.ru;2 MIPT, Moscow, Russia; vicsol@mail.ru

The surface barrier discharge (SBD) is a discharge between the flat elec-trodes divided by a layer of dielectric which interferes with chain short circuitbetween electrodes and to emergence of a arch discharge. Intensive researchof the surface barrier discharge (SBD) began about 10 years ago in connectionwith prospects of its use for management of laminar and turbulent transitionand the provision of zones of a separation of a current in aerodynamic appen-dices [1-4]. Recently the area of intended application of it like the dischargeextended. The discharge serves as means of initialization of burning of fuels atthe expense of an operating time in a zone of the discharge of active radicals andmetastable molecules in the electronic raised conditions [5,6]. Any applicationsof the discharge, naturally, demand detailed knowledge of physical processesof its development. The physical model of the phenomenon is constructed onthe basis of the analysis of all effects defining dynamics of development of thedischarge. Movement of charges is described by system of the equations of thetransfer, Poisson added with the equation for electric field and its potential [7]Complexity of numerical modeling of the considered phenomenon is connectedwith existence of boundary layers at an electrode and at a dielectric surface,and also mobile narrow zones of ionization. It should be noted that the self-coordinated electric fields are very sensitive to spatial redistribution of charges.Poisson’s equation needs to be solved on each step on time. The last circum-stance does calculations extremely labor-consuming. The developed numericalmethods are presented in the report for computer modeling of this phenomenon.Testing of the received results is held by comparison with available experimentaldata.

The authors were supported by the Program of basic researches N3 OMNRussian Academies of Sciences).

References

1. J.R. Roth , D.M Sherman, S.P. Wilkinson . “Electrohydrodynamic Flow Con-trol with a Glow Discharge Surface Plasma ,” AIAA Journal , 1, No. 38,1166–1172 (2000)

103

2. G.Artana, J.DAdamo, L.Leger, E.Moreau, G.Touchard. “Flow Control withElectrohydrodynamic Actuators ,” AIAA Journal , 1, No. 40 1773-1779(2002)

3. E.Moreau. “Airflow control by non-thermal plasma actuators ,” J. Phys. D:Appl. Phys., 1, No. 40 605-636(2007)

4. T.C.Corke, M.L.Post, D.M. Orlov. “Single dielectric barrier discharge plasmaenhanced aerodynamics: physics, modeling and applications ,” Exp.Fluids ,1, No. 46 1-29 (2009).

5. F.O.Thomas, T.C.Corke, M.Iqbal, A.Kozlov, D.Schatzman. “Optimization ofDielectric Barrier Discharge Plasma Actuators for Active Aerodynamic FlowControl ,” AIAA Journal , 1, No. 47 2169-2178(2009).

6. A.Yu.Starikovskii, N A.A.ikipelov, M.M.Nudnova, D.V.Roupassov. “SDBDplasma actuator with nanosecond pulse-periodic discharge ,”Plasma SourcesSci. Technol. , 1, No.18 034015(2009).

7. V.R.Soloviev, V.M.Krivtsov. “Surface barrier discharge modeling for aerody-namic applications,”J. Phys. D: Appl. Phys , 1, No. 42 125208(2009).

104

The problem of scheduling a freight train as aproblem of integer programming.

Alexander Lazarev1, Elena Musatova2, Nail Husnullin3

1 V.A.Trapeznikov Institute of Control Sciences RAS, Moscow, Russia;M.V.Lomonosov Moscow State University;

National Research University Higher School of Economics; jobmath@mail.ru2 Institute of Control Sciences RAS, Moscow, Russia; nekolyap@mail.ru

3 Institute of Control Sciences RAS, Moscow, Russia; nhusnullin@gmail.com

The central question that motives this paper is the problem of making up afreight train and the routes on the railway. It is necessary from the set of ordersavailable at the stations to determine time-scheduling and destination routingby railways in order to minimize the total completion time. In this paper it wassuggested formulation of this problem by applying integer programming.

Moreover, In this paper it was shown that depending on the situation, thetask can be formulated in different ways: the number of train can be fixed orleave arbitary; change objective function, routes, train schedules and so on. Inthe all situation the task is a problem of integer programming and for whichexists the number of exact or approximation approaches. It’s important toimphasize that the effectiveness of the problem solution it depends on analysis ofthe structure of the constraint set and select an appropriate method of solution.

References

1. Lazarev A.A., Musatova E.G., Gafarov E.R., Kvaratskheliya A.G. Theory ofScheduling. The tasks of railway planning. – M.: ICS RAS, 2012. – p.92

2. Lazarev A.A., Musatova E.G., Gafarov E.R., Kvaratskheliya A.G. Theoryof Scheduling. The tasks of transport systems management. – M.: PhysicsDepartment of M.V.Lomonosov Moscow State University, 2012. – p.160

3. A. Caprara, L. Galli, P. Toth. Solution of the Train Platforming ProblemTransportation Science. 2011. - 45 (2), P. 246-257.

105

On improvement of local algorithms for discreteoptimization problems

Daria Lemtyuzhnikova1, Aleksander Sviridenko2, Oleg Shcherbina3

1 Tavrian National University, Simferopol, Ukraine; darabbt@gmail.com2 Tavrian National University, Simferopol, Ukraine;

oleks.sviridenko@gmail.com3 Tavrian National University, Simferopol, Ukraine;

oleg.shcherbina@univie.ac.at

The development of the decomposition methods in discrete optimization(DO) is now very timely. Usually, DO problems from applications have a specialstructure, and the matrices of constraints for large scale problems have a lotof zero elements (sparse matrices). The nonzero elements of the matrix oftengroup in a graph decomposition schemes and logic-based approaches to discreteoptimization problems limited number of blocks. The block form of many DOproblems is usually caused by the weak connectedness of subsystems of realsystems. Among the decomposition approaches appropriate for solving sparseDO problems we mention local elimination algorithms (LEA) [1], [2] using thespecial block matrix structure of constraints which exploit the sparsity in theinteraction graph of a DOP and allow to compute a solution in consecutivestages, where each stage uses results from previous ones. LEA is describedin detail in [2]. The main drawback of LEA appears for the large size of theseparator, which separates the various blocks because a volume of enumerationis exponential on size of the separator.

To reduce the amount of enumeration, we propose to use relaxations in acomputational scheme of LEA, as it has successfully done in branch and boundalgorithms. Thus, to solve the DO problem in block, we’re using objective func-tion values of relaxed problems instead of actual objective function values (e.g.the linear relaxation, when instead of the integer linear programming problem,one considers the corresponding linear programming relaxation, knapsack re-laxation, where instead of the original constraints of the problem we considera surrogate constraint, which is a linear combination of the original constraintsetc.).

Preliminary experiments showed a significant reduction of run time whenusing relaxations. The use of different relaxation techniques in local eliminationalgorithms is the subject of further research.

Even on contemporary serial computers solving combinatorial optimizationproblems is somewhat limited from a computational point of view. Parallelprocessing computers could greatly reduce the computation time for solving

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large scale DO problems since there are a number of parallel operations thatoccur in solving DO problem. Due to the high computational requirements andinherent parallel nature of DO search techniques, there has been a great dealof interest in the development of parallel DO search methods since the dawn ofparallel computing.

The main disadvantage of the LEAs, its great computational complexity forDO problems with high size of separators, can be reduced with parallel process-ing on clusters of workstations. In order to reduce the amount of enumeration,postoptimality analysis can be used, since DO problems in the package, thatcorresponds to a block, differ in right-hand sides. However, our experimentshowed a slight decrease in run time when using postoptimality analysis. An-other possibility to cope with the challenges posed by the large size separatorsis to develop approximate versions of the local algorithm, which does not com-pletely enumerate all possible values of the variables included in the separator,but only partially. Thus, the only one way to implement and use exact localelimination algorithms for sparse DO problems with large separators is usingof parallelization. Parallelization in DO can help to cope with this drawbackusing the possibility of parallel solving of DO problems in blocks. There aretwo types of parallelism available in LEA:

1) Parallelism of type 1 introduces parallelism when performing the oper-ations on generated subproblems (in blocks). It consists of solving each DOsubproblem in parallel for each block to accelerate the execution.

2) Parallelism of type 2 consists of processing the elimination tree in parallelby performing operations on several subproblems simultaneously. Independentbranches of the elimination tree can be processed in parallel, and we refer tothis as type 2 parallelism or tree parallelism.

References

1. Yu.I. Zhuravlev. Selected Works, Magistr, Moscow, (1998). (in Russian).

2. O.A. Shcherbina. “Local elimination algorithms for solving sparse discreteproblems.” Computational Mathematics and Mathematical Physics, 48, 152–167 (2008).

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Pareto Frontier Visualization for Nonlinear Modelsand its Use in Multiobjective Search for Efficient

Therapy of HIV Virus Infection

Alexander Lotov1

1 Russian Academy of Sciences, Dorodnicyn Computing Centre, Moscow,Russia; avlotov@gmail.com

Visualization of the Pareto frontier is an effective decision support tool inthe case of multiple criteria (objectives). The Pareto frontier methods startwith the approximating of the Pareto frontier. Then, the decision maker isinformed about the Pareto frontier. In problems with more than 2 criteria, thedecision maker needs to study the multi-dimensional Pareto frontier and relatedobjective tradeoffs.

In this paper, an approach to Pareto frontier visualization is used, whichis based on approximating the Edgeworth-Pareto Hull (EPH) of the feasiblecriterion set. To visualize the Pareto frontier, collections of bi-criterion slices ofthe EPH are displayed in a dialogue with the decision maker. The approach wassuccessfully used for constructing efficient strategies in economic, environmentaland other problems (see [1]).

In this study, visualization of the Pareto frontier is applied in the process ofsearching for the efficient therapy of the HIV virus infection. A non-linear math-ematical model of the virus dynamics is used. A therapy strategy is constructedthat balances the conflicting requirements imposed on the therapy process (seefor details in [2]).

The authors were partially supported by the Russian Foundation for BasicResearch (projects no. 13-01-00235 and no. 13-01-00779) and by the Programsfor Fundamental Research of RAS Pi-15 and Pi-18.

References

1. A.V. Lotov, V.A. Bushenkov, G.K.Kamenev. Interactive Decision Maps. Ap-proximation and Visualization of Pareto Frontier, Kluwer, Boston (2004).

2. A.V. Lotov, A.S. Bratus, N.S. Gorbun. “Pareto frontier visualization in multi-criteria search for efficient therapy strategies: HIV infection example,” Rus-sian Journal of Numerical Analysis and Mathematical Modelling, 27, No. 5,441–458 (2012).

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On firm investment policy model accounting foremployees learning

Maria Makarova1, Nicholas Olenev2

1 Moscow Institute of Physics and Technology, Moscow, Russia;makarova.mar@gmail.com

2Dorodnicyn Computing Centre of RAS, Moscow, Russia; nolenev@yahoo.com

In the article we build a modification of the model, developed in the articleby N.N. Olenev, I.G. Pospelov. The main assumption is that capacity of afirm depends on the time non-monotonically, and it also depends on cumulativeproduction of the firm (Y). We use the assumption that number of employeesof the firm is constant.

m(t) = F (Y (t), t)e−µt, (1)

Y (t) =

∫ t

0

y(t)dt, (2)

y(t) = m(t)θ(t), (3)

Where θ(t) ∈ [0, 1] - is controlling parameter. The function F (Y (t), t) is in-creasing and concave and stands for staff learning. The firm solves the problemof maximizing profits (p is price of product, s is cost of labor, λ is standardlabor laboriousness)

P =

∫ t

0

(pF (Y (t), t)e−µtθ − λs)tdt→ max, (4)

Y ′(t) = F (Y (t), t)e−µtθ, (5)

applying necessary conditions of the extreme we get the general formula forthe decision

θ =

1, t ∈ [t0;T ]

0, ohterwise(6)

Where t0 is time of production beginning ad is defined by the equation

pY ′t0(t0, T )− λs = 0, (7)

T is the time when production is no more profitable

pF (Y (t0, T ), t)e−µt − λs = 0, (8)

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Consider sustainable economic growth when p and s are constant and num-ber of firms created changes as

J(t) = J0eγt, (9)

Then total production rate is

ysum(t) = J0eγt

∫ T

t0

e−γτF (Y (t0, T ), τ )e−µτ ≡ J(t)f(x, γ), (10)

Number of employed is

N(t) =J0e

γtλ

γ(e−γt0 − e−γT ) ≡ J(t)x, (11)

Assuming eγt ≈ eγt0 and using (7), (8), we get an approximate equation

∂f(x, γ)

∂x≈ sλ

p(12)

Production per capita is

w =f(x, γ) + pb

x(13)

Where b is capital intensity (number of production items required to createone firm)

By maximizing it we get an analogue of Solows rule

f(x, γ) = x∂f

x+ b ≈ xs/p+ b (14)

References

1. N.N. Olenev, A.A. Petrov, I.G. Pospelov. ”Model of Change Processes ofProduction Capacity and Production Function of Industry ,” MathematicalModelling: Processes in Complex Economic and Ecologic Systems , 1, 46–60(1986).

2. N.N. Olenev, I.G. Pospelov. ”The Model of Investment Policy in MarketEconomy ,” Mathematical Modelling: Processes in Complex Economic andEcologic Systems, 1, 163–173 (1986).

3. S.V. Konyagin. Variational calculus and optimal control, MSU, Moscow, 2005

110

The method of control models in problems oftracking of solutions of differential equations

Vyacheslav Maksimov1

1 Institute of Mathematics and Mechanics, Ural Branch, Acad. Sci. of Russia,Ekaterinburg, Russia; maksimov@imm.uran.ru

The problems of tracking of solutions of dynamical systems described bydifferent classes of equations: ordinary differential equations of the form

x(t) = f(t, x(t)) +Bu(t), t ∈ T = [t0, ϑ],

differential equations with distributed parameters of parabolic type

w(t) + Aw(t) = Bv(t) + f(t), t ∈ T = [t0, ϑ], w(t0) = w0,

and nonlinear systems describing the process of solidification (a phase fieldequations)

∂

∂tψ + l

∂

∂tϕ = k∆Lψ + u Ω× (0, ϑ], ϑ = const < +∞,

τ∂

∂tϕ = ξ2∆Lϕ+ g(ϕ) + ψ,

are under investigation. Solving algorithms that are stable with respect to infor-mational noises and computational errors and operate in the real time mode, aredesigned. These algorithms are based on the method of control models from thetheory of ill-posed problems and on the well-known in the theory of positionaldifferential games control principle of extremal shift. Inaccurate measurementsof current phase states of systems play the role of input data for the algorithms,which provide some values approximating real (but unknown) characteristics.The basic elements of the algorithms are represented by stabilization procedures(functioning by the feedback principle) for appropriate Lyapunov functionals.

The authors were supported by the Russian Foundation for Basic Research(project no. 13-01-00110).

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Parallel Method of Nonuniform Coverings

V.U. Malkova

Dorodnicyn Computing Centre of RAS, Moscow, Russia; vmalkova@yandex.ru

Modern multiprocessor computers considerably extend the capability ofsolving global optimization problems by using parallel algorithms. A paral-lel method for the global optimization is developed on the basis of the methodof nonuniform coverings proposed by Yu.G. Evtushenko [1]. If the Lipschitzconstant is known, the method of nonuniform coverings yields a guaranteedsolution; however, only problems of low dimension can be solved in practice.Numerical results show that the efficiency of this method considerably dependson the Lipschitz constant, which is usually not known a priori; therefore, toohigh upper bounds are used, which considerably complicates the computations.At the same time, there are domains in many problems in which the Lipschitzconstant is not large and the use of the maximal constant is not reasonable. Weconsider the minimization of functions of three types: functions satisfying theLipschitz condition; functions whose gradient satisfies the Lipschitz condition;functions whose Hessian satisfies the Lipschitz condition. For these types offunctions, the general scheme of the nonuniform covering method is elaboratedwith the use of local optimization.

The main idea behind the proposed parallel algorithm is to perform itera-tions concurrently on several processors that periodically exchange the incum-bent minima; the search domains are periodically redistributed between theprocessors in the course of the computations. To concurrently perform the it-erations, one needs a method for distributing the parallelepipeds between theprocessors. Assume that there are p processors and there is an initial sequenceBm = P1, ..., Pm of the parallelepipeds Pi belonging to P . The distribution ofthe work between the processors depends on the unit of work of the algorithm.As such a unit, we use the computational work needed to perform an iteration ofthe sequential algorithm Q times. Assume that, in addition to the p processorsmentioned above, which will be called slaves, there is an additional processorcalled the master. Let us distribute the parallelepipeds in the initial collectionBm among the slave processors. When performing the iterations, each slaveprocessor maintains its own pool of parallelepipeds to be examined. We saythat a processor finished its portion of the work if it performed Q iterationsof the sequential algorithm (performed Q bisections) or if its individual pool isempty. The processor that finished its work sends to the master the followingdata: the number of parallelepipeds in its pool, the minimal bound y for each ofthem, and the incumbent minimum of f obtained in the course of the calcula-

112

tions. This processor is placed in the waiting queue maintained by the master.A waiting processor is said to be free if its individual pool is empty.

The master completes the execution of the algorithm when all the slaveprocessors are free. In the proposed algorithm, the frequency of the communi-cations between the slave processors and the master depends on the parameterQ. As soon as a free processor is detected, the work is redistributed between theprocessors by passing parallelepipeds. In such a scheme, the processors executethe algorithm asynchronously exchanging information only with the master.

This method is implemented in C-MPI for the global minimization of func-tions whose gradient satisfies the Lipschitz condition. The performance of thealgorithm is demonstrated using the calculation of the structure of a proteinmolecule as an example.

The variants of the method of nonuniform coverings make it possible toglobally optimize Lipschitzian functions. The requirements that the Lipschitzconstant is known and invariable in the entire search domain are removed. Thedifferentiability condition of the function is introduced. Due to this condition,the lower bounds of the function values in the current search domains are con-siderably improved, which reduces the computation time by accelerating thepruning of ”unpromising” parallelepipeds. The proposed parallel global opti-mization method can be used to solve multicriteria optimization problems.

This work was supported by the Programs of Fundamental Research ofRussian Academy of Sciences P-18, and by the Leading Scientific Schools Grantno.5264.2012.1.

References

1. Yu.G. Evtushenko. “A Numerical Method for Finding Best Guaranteed Esti-mates”, Zh. Vychisl. Mat. Mat. Fiz. 12 (1), 89-104 (1972)

2. Yu.G. Evtushenko, V.U. Malkova, and A.A. Stanevichyus. “Parallel GlobalOptimization of Functions of Several Variables”, Computation Mathematicsand Mathematical Phisics, Vol. 49, No. 2, 246 – 260 (2009)

113

On optimal control of motion of transport systems

Olga Masina

Yelets State University after I.A. Bunin, Yelets, Russia; olga121@inbox.ru

Numerous works (see, for example, [1]–[3]) are devoted to questions of opti-mization of dynamics of controlled systems. In the present work we consider thequestions of motion control of the transport systems described by the differen-tial equations in contingencies that allows to carry out the precised analysis ofstability of the nominal motion determined by criterion of an optimality. Themultivalence in the right parts of the differential equations arises in view ofresistance of the rarefied environment which as a first approximation is not con-sidered. We use the conditions of existence and stability of differential inclusions[4]–[6] for studying of controlled systems. Results of this work are continuationand developement of works [7], [8].

The problem of optimal control in which the controlled object moves in thevertical plane from the initial point in final with intermediate achievement ofthe set height is considered.

At the first stage motion occurs to constant vector traction (p, q); at thesecond stage with constant traction (−r, s) and, unlike researches of other au-thors [2], [3], values of positive parameters p, q, r, s get out of intervals (p1, p2),(q1, q2), (r1, r2), (s1, s2) respectively. The object is affected also by the forceof gravity with constant acceleration g, and resistance of the environment isneglected. Object movements at the first and second stages are described bydifferential inclusions of the form

mx′′ ∈ p,my′′ ∈ q −mg,

mx′′ ∈ −r,my′′ ∈ s−mg.

Motion will be admissible if coordinates x, y of object are satisfy the condi-tions: x(t) grows from 0 to l, y(t) increases at the first stage up to the heightof h and y(t) decreases to zero at the second stage. Thus at the time landing t2is when y(t2) = 0, will obtain x(t2) = l.

The problem of optimal control lies in selection of the parameters p, q,r, s and values t1, t2 so that object motion from the initial point O(0, 0) infinal L(l, 0) with intermediate achievement of height h was carried out with theminimum fuel consumption. As criterion of quality we have the condition:

∀ p ∈ (p1, p2), ∀ q ∈ (q1, q2), ∀ r ∈ (r1, r2), ∀ s ∈ (s1, s2),

114

t1∫

0

(p+ q)dt+

t2∫

t1

(r + s)dt→ min .

In the first step the optimal control for single-valued parameters is defined.In a case when parameters are defined ambiguously, motion correction by meansof sensors for return to an optimal route is required. At constant parameters thecorresponding calculations were carried out in system of computer mathematicsof Mathematica 4.1. We propose the algorithm for a control system on objectmoving to the purpose in three-dimensional space when the purpose movesrandomly with a limited speed on the plane. This algorithm can be realized inthe form of the computer program.

The author is suppoted by the Russian Foundation for Basic Research(project no. 13-08-00710-a).

References

1. V.I. Blagodatskich. Introduction in optimal control (the linear theory), Higherschool, Moscow (2001).

2. J. Varga. Optimal control of the differential and functional equations, Trans-lated from English, Moscow (1977).

3. L.S. Pontryagin, V.G. Boltyansky, R.V. Gamkremidze, E.F.Mishchenko. Mathematical theory of optimal processes, Science, Moscow(1961).

4. A.A. Shestakov. Generalized direct Lyapunov’s method for systems with dis-tributed parameters, URSS, Moscow (2007).

5. Yu.N. Merenkov. Stability-like properties of differential inclusions, fuzzy andstochastic differential equations, Russian University of Peoples’ Friendship,Moscow (2000).

6. O.N. Masina. ”On the existence of solutions of differential inclusions”, Dif-ferential Equations, 44, No. 6, 872–875 (2008).

7. O.N. Masina. ”Questions of motion control of transport systems”, Transport:science, technique, control, No. 12, 10–12 (2006).

8. O.V. Druzhinina, O.N. Masina. Methods of stability research and controlla-bility of fuzzy and stochastic dynamic systems, Dorodnicyn Computing Cen-ter of RAS, Moscow (2009).

115

An optimal control problem with state and mixedconstraints

Arsalan Mizhidon1, Klara Mizhidon1

1 East Siberia State University of Technology and Management, Ulan-Ude,Russia; miarsdu@mail.ru

Consider the linear control system

x = Ax(t) +Bu(t) + f(t), x(0) = x0, (1)

where x – n-dimensional vector of state variables; u – r-dimensional controlvector; A – constant (n × n)-matrix; B – constant (n × r)-matrix; f(t) – n-dimensional vector of external disturbances. The vector function f(t) is limitedwhen t ≥ 0.

The system is controlled on a time interval [0, T ]. In this case admissiblecontrols u(t) are continuously differentiable functions.

Normal operation of the system involves at each time t ∈ [0, T ] satisfactionof the state constraints

x′(t)Qıx(t) ≤ 1, (ı = 1, 2, . . . , s) (2)

and mixed constraints

x′(t)Cx(t) + x′(t)Pu(t) + u′(t)Du(t) ≤ 1, ( = 1, 2, . . . , p), (3)

where P – (n× r)-matrices; Qı, C – non-negative definite, symmetric (n×n)-matrices; D – non-negative definite, symmetric (r× r)-matrices; bar ()′ – hereand below the operation of transposition. Matrices P, C, D are such thatmatrices

G =

(C

12P

12P ′ D

)

are positive definite matrices.It is required for a given perturbation to find such control u∗(t) that at

each time t ∈ [0, T ] the state (2) and mixed (3) constraints are satisfied on thetrajectories of the system (1) and minimum of the following functional

J(u) =

T∫

0

u′(t)Ru(t)dt (4)

is attained, where R – positive definite, symmetric (r×r)-matrix. The functional(4) describes expenditure of energy.

116

To solve the problem it is proposed to consider the auxiliary optimal controlproblem with quadratic performance measure [1] with matrices depending onweight coefficients. The choice of these coefficients provides satisfaction of thestate (2) and mixed (3) constraints with a minimum of performance measure(4).

The auxiliary optimal control problem:

x = Ax(t) +Bu(t) + f(t), x(0) = x0,

x(t), f(t) ∈ En, u(t) ∈ Er, t ∈ [0, T ],

J(u(·)) = 12

T∫0

(x′Q(α, β)x+ x′P (β)u+ u′R(β)u) dt→ min,

(5)

where Q(α, β), P (β) and R(β) – matrices respectively of dimension (n × n),(n× r) and (r × r) are defined as follows

Q(α, β) = α1Q1 + α2Q2 + . . .+ αsQs + β1C1 + β2C2 + . . .+ βpCp,

P (β) = β1P1 + β2P2 + . . .+ βpCp,

R(β) = R + β1D1 + β2D2 + . . .+ βpDp.

Here αı and βı are weight coefficients.The auxiliary optimal control problem reduces to solving the boundary prob-

lem of the maximum principle.The construction of algorithmic support of the boundary problem solution

and an iterative procedure for selecting weights is based on a representation ofthe fundamental matrix in the form of an exponential matrix of the system,arranged in a matrix row.

The authors were supported by the Russian Foundation for Basic Research(project no. 12-08-00309).

References

1. A.D. Mizhidon “On a problem of analytic design of an optimal controller”,Automation and Remote Control, 72, No. 11, 2315–2327 (2011).

117

Inverse problems of output modeling in theconditions of globalization

Evgeny Molchanov1, Alexander Shananin2

1 MIPT, Moscow, Russia; molch6464@phystech.edu2 MIPT, CCAS, Moscow, Russia; alexshan@ysndex.ru

Globalization and the integration of Russia into the world economy areamong the trends that significantly affect the economic situation of Russia now.Over the past decade, the globalization process on a global scale has led tosignificant changes in the goods substitution elasticity. In this regard, a numberof new problems arises requiring modification of the basic models of economictheory and the study of inverse problems, which are necessary for processing ofeconomic statistics in the new environment.

The need to reflect the competitive process of domestic goods and their im-ported counterparts requires not only the modification of the model, but also thenew methods of statistical computing. For example, the traditional techniquesdeveloping in a planned economy based on input-output analysis. However, as aresult of the import substitution process share of domestic manufacturing indus-try goods varies greatly depending on the economic environment, and as a result- the constancy manufacturing costs hypothesis does not hold. At a time whenthe Central Bank of Russia to maintain a stable exchange rate against foreigncurrencies, imported goods displace domestic because of the higher rate of infla-tion in the Russian domestic market compared with the world one. When therising financial difficulties lead to a devaluation of the ruble reverse process ofimport substitution by domestic starts. Therefore, when describing the agentsit is necessary to model their behavior, in particular to describe the process ofchoosing between domestic and imported goods-resources within the productionprocess. It is planned to use the Houthekker-Johansen modified model, whichtakes into account the substitution of imported and domestic goods at the microlevel. To estimate the elasticity of substitution between imported and domesticgoods at the micro level will be investigated corresponding inverse problems ofintegral geometry.

We develop methods witch take into account the substitution of the currentuse production factors from the producers in the construction of models andprocessing statistics. Emerging inverse problems belong to a class of problemson the inversion of the generalized Radon transform for a parameterized familyof hypersurfaces.

The problem is as follows: to find the elasticity ε if there exists a nonnegative

118

measure µ() in R2+, µ(R

2+) <∞ such that

yt(pt1, pt2, p

t0) =

∫

R2+

θ(pt0 − q(pt1x1, p

t2x2)

)µ(d(x)),

wherept1, p

t2, t = 1, . . . , T – price time series of the imported and domestic current

use production factors;pt0, y

t, t = 1, . . . , T – price and quantity time series of output goods;

q(p1x1, p2x2) =(p1x

−ρ1 + p2x

−ρ2

)− 1ρ – CES function with the elasticity ρ, ρ =

− ε1+ε

;θ() – Heaviside step function.Compared with the problems encountered in computer tomography, unstruc-

tured grids information is available in the processing of economic productionstatistics. Investigation of arising moment problems is related with the studyof combinatorial problems on wiring deformations and dual objects - tylings.

The authors were supported by the Russian Foundation for Basic Research(projects no. 12-07-31231, 11-07-00162) and the The Ministry of Education andScience of Russia (project no. 14.A18.21.0866)

References

1. A.A. Shananin. “The generalized model of a pure industry” (in Russian),Mathematic Modeling, 9, No 9, (1997).

2. A.A. Shananin. “The investigation of the generalized model of a a pure in-dustry” (in Russian), Mathematic Modeling, 9, No 10, (1997).

3. A.A. Shananin. “Non-parametric method for the analysis of industry techno-logical structure.” (in Russian), Mathematic Modeling, 9, No 11, (1999).

119

Quasi-time-optimal controls for wheeled robot

Yury Morozov1

1 ICC RAS, Moscow, Russia; tot1983@inbox.ru

Consider a planar motion of a mobile robot. The kinematics of mobile robotcan be described as

x = y, y = φ3(y)u+ z(ξ), z = φ(y)2c(ξ)

(1− c(ξ)x)), φ(y) =

√1 + y2, (1)

where dot denotes derivation with respect to the independent variable ξ, x ∈ Ris a distance to the given curvilinear path, y ∈ R is a angle deviation from thegiven path, c(ξ) < C is a curvature of the given path, u denotes a control.

P r o b l e m 1. The goal of this paper is to determine control inputsu(x, y, C) subject to |u(x, y, C)| < U for a mobile robot to place a target pointon robot’s platform to a given curvilinear path in finite-time and to stabilizemotion along this path.

We will consider z(ξ) = φ(y)2c(ξ)/(1− c(ξ)x)) as a parametric perturbation[1]. To solve Problem 1, we apply quasi-time-optimal control designed for astraight target path (see, e.g., [2])

u = −Usign(S), S = Uλx+(1− φ(y)−1

)sign(y), Uλ =

U

1 + λ, (2)

whereλ > λ∗, δ < U, (3)

where λ∗ = δ/(U − δ), δ = maxy(δc(y)) = δc(0), δc(y) = C/(1 − Cx∗)/φ(y), δca boundary of the parametric perturbation, x∗ is a given boundary of variablex.

Lemma 1. Let the system (1) be closed by (2) and the inequality (3) be hold,than system (1) is local stabilized in finite time by the state-feedback control (2)and there is attracting sliding mode [3] along curve s(x, y) = (x, y)T : S(x, y) =0, |x| ≤ x∗.

120

S(x,y)=0F+

F-

S>0

S<0

Fig. 1. Attraction domain, switching curve and phase pathes.

Using the worst parametric perturbation, we construct new switching curve

and control SC = xUλ + ln(|1− C|x||)sign(x) +(1− φ(y)−1

)sign(y),

u = −sign(SC(x, y)). Let us the parameter λ∗ be calculated as λ∗ = argx<0SC(−xc, 0) =0 = Uxc/ ln(1− Cxc)− 1 to get sliding mode.

Fig. 2. Attraction domains and switching curves

This work was supported by the Russian Foundation for Basic Research(RFBR) grant no. 13-01-00347. and and the Department of Power Engineering,Mechanical Engineering, Mechanics and Control Process of Russian Academy ofSciences (Program no. 1 “Scientific foundations of robotics and mechatronics”).

References

1. A. Balluchi et al. “Path tracking control for Dubin’s cars,” 3123–3128 (1996).

1. Yu. V. Morozov “Quazi-Time-Optimal Control of wheeled vehicle,” Proc.IEEE International Conference Robotics and Automation, 73–74 (2012).

3. A. F. Philippov “Differential equations with discontinuous right-handside.,”Nauka, M., 1985.

121

Reverse and optimisation problems in creep process

Evgenii Murashkin1

1 Institute for Automation and Control Processes Far Eastern Branch RussianAcademy of Sciences, Vladivostok, Russia; murashkin@dvo.ru

In the design of ship hulls and aircraft are widely used panels and profiles ofhardly-deformed aluminum alloys at normal temperatures. Traditional methodsof formation of structural elements of such alloys often leads to the appearanceof the plastic breaks, cracks and other damage. In this regard, effective way isformation under high temperature and low deformation under creep conditions.The use of such process ensures the production of parts with high accuracy,which reduces the complexity of assembly and welding, and also reduces theformation of effort to improve the residual service life and quality of construc-tion. At the same time it is necessary to solve problems of modeling the processof formation, the timing and strength of deformation regimes, the determina-tion of the geometry of the light snap of an elastic response after removal of theloading effort. In this case there is a necessity of calculation of such processesin the model with finite irreversible deformations and complicated rheologicalproperties of materials. Consideration be carried out in the model of finiteelastoplastic creep deformations [1].

We formulate the optimal deformation problem in the creep [2]: we arenecessary to define a way of deformation of elasto-creeping material within aspecified period of time to the final point of unloading to obtain the specifiedvalues of residual strains with minimal damage parameter.

The authors were supported by the Grant of the President of the Rus-sian Federation (project no. MK-776.2012.1) and the Grant of RFBR (projectno. mol a ved 12-01-33064).

References

1. Bazhin A.A., Murashkin E.V. Creep and Stress Relaxation in the Vicinityof a Micropore under the Conditions of Hydrostatic Loading and Unloading.Doklady Physics, Pleiades Publishing, Ltd. ISSN 1028-3358. 2012, Vol. 57,No. 8, pp. 294-296.

2. I.Yu. Tsvelodub. A class of inverse creep theory problems. Journal of AppliedMechanics and Technical Physics 30, 320–329, 1989.

122

Modeling and optimization of ion-beam etchingprocess II

L.A.Muravey, V.M.Petrov, A.M.Romanenkov

MATI-RSTU named after K. Tsiolkovsky, Moscow, Russia; pm@mati.ru

The ion-beam etching (IBE) is one of the types of the semiconductor ma-terials surface treatment. The purpose of the work is the IBE process controlfor maximum approximation of the surface being etched to the step form. Theion-beam etching process running can be affected by changing the direction ofthe ion beams with time. The evolution of the surface, having an arbitrary formduring the ion bombardment process, is described by the equation

ϕt(t, x) + ϑ(θ)√

1 + ϕ2x(t, x) = 0 (1)

where ϕ(t, x)-is the height of the etched form at the time moment t in position x,ϑ(θ)-is the rate of the material ion sputtering, dependant on the angle θ, formedby the ion incidence angle and the normal to the surface being sputtered. Theetching equation (1) has the same form both for a plane case and for the rotationin three-dimensional space (see [1]).

The case, when the sample is rotating with the angular velocity ω will beconsidered. In this case it has been proved, that the angle θ satisfies the equation

cos θ =cosα(t)− ϕx(x, t) sinα(t) sinωt√

1 + ϕ2x(x, t)

(2),

where α(t) is the angle between the direction of ion beam incidence and axisz. Well consider α(t) as the control, upon which the natural restrictions areimposed:

0 ≤ α(t) ≤ αmax (3)

It will be assumed, that the etching process terminates, when the working layeris etched to the depth H. Well be interested in the rate of the point x0 movementto the left, where x0 is the boundary point between the protective mask andthe working layer. It has been proved (see [1]), that x0 satisfies the followingdifferential equation:

dx0(t)

dt= ϑ(θ(t))

√1 + ( ∂ϕ

∂x)2 |ϕ=0

∂ϕ∂x

|ϕ=0

(4)

123

The optimal control problem is as follows. To determine the functions ϕ(t, x)and α(t), satisfying the equations (1), (4), the restrictions (3), the specifiedinitial conditions and the condition

ϕ(T, 1) = −H, (5)

so that the function x0(t) at the terminal time moment t = T acquired themaximum value, i.e.

J = x0(T ) → max (6)

Thus, our problem is that one with the non-fixed time T. In the work [1] it hasbeen shown how the turn to the problem of optimal control with fixed time canbe executed. The use of the singular variations has enabled us to obtain themaximum principle in the form:

(ϑ(θ(t, x0(t)) + α)

ϑ(α)

)→ max

α

The appropriate calculations have been performed.

References

1. L.A.Muravey, V.M.Petrov, A.M.Romanenkov. Modeling and optimization ofion-beam etching process//Proceedings. III International conference on op-timization methods and applications (OPTIMA-2012). Costa da Caparica,Portugal, September 2012. ISBN 978-5-91601-051-0.

124

Inscribing a cube of maximum volume in a region ofacceptability for optimal parameters sizing

Dmitry Nazarov

Institute of Automation and Control Processes FEB RAS, Vladivostok,Russia; nazardim@iacp.dvo.ru

The task of optimal parameters sizing at engineering systems design usuallyassociated with feasible paramer region exploration or contruction of a regionof acceptability (RA). RA is a domain comprised of parameter vectors whichyield proper system performance (1).

Dx = x ∈ Rn | ymin ≤ y(x) ≤ ymax, (1)

where, y(x) is an engineering system model which is usually given in implicitform, and ymin and ymax are the system performances constraints. The diffi-culty of RA construction consists in high dimension of parameter space, incom-plete prior information and only pointwise exploration of parameter space withsystem performances calculation [1].

RA facilitates the procedure of nominal parameters sizing with the respectof reliability requirements. This procedure requires patterns of parameter de-viations and distribution laws which allow to predict system gradual failure.In case of lack of this information the deterministic criterion of performancereserve that consists in finding such a vector xo ∈ Dx that is the most distantfrom the border of Dx. This point is a centre of a convex and symmetricalfugure inscribed into Dx (e.g. n-dimensional sphere). Formal statement of thecriteria is given in (2)

maxx∈Dx

minrV (x, r), (2)

where function V (x, r) is a volume of inscribed figure. The volume dependson the center point x ∈ Dx and the distance r to the border of Dx. This figureprovides the minimum of performance reserve in the worst case of parameterdeviation (Fig. 1).

125

x1

x2

0

D x

xo1

xo

xo3

xo2

rmin1

rmin3

rmin2

rmino

Fig. 1

In this work, a grid representation of a RA is used [2]. This representation isbased on a n-dimensional region approximation with a discrete set Bgx of elemen-tary parallelepipeds (boxes) described with n-dimensional grid and determinedwith indices (k1, k2, . . . , kn). For this representation of a RA, the criterion (2)is reduced to inscribing a symmetrical figure consisting of elementary boxesthat belong to RA representation B

+x ⊆ B

gx and central element with indeces

(kc1, kc2, . . . , k

cn). One of such figures can be a r-cube:

Brc = ek1, k2, ..., kn ∈ B

gx | kci − r ≤ ki ≤ kci + r ∀i = 1, 2, . . . , n, (3)

As the result a group of optimal elementary boxes can be found. In this casethe values of optimal parameters should be calculated as geometrical average ofcentral points of the elementary boxes of their every connected set [3].

The work is partially supported by the OEMMΠY RAS No.14 programme(project no. 12-I-OEMMΠY-01).

References

1. R. Spence, R.S. Soin. Tolerance Design of Electronic Circuits, World Scientific(1997).

2. O.V. Abramov, D.A. Nazarov. “Regions of Acceptability in Reliability De-sign,” Reliability: Theory & Applications , 7, No. 3(26), 43–49 (2012).

3. O.V. Abramov, Y.V. Katueva, D.A. Nazarov “Reliability-Directed Dis-tributed Computer-Aided Design System,” Proc. of the IEEE InternationalConference on Industrial Engineering and Engineering Management - Singa-pore, 1171–1175 (2007).

126

Distributed Optimization in Networked Systems

Angelia Nedic1

1 University of Illinois, Urbana, USA; angelia@illinois.edu

Recent advances in wired and wireless technology necessitate the develop-ment of theory, models and tools to cope with new challenges posed by large-scale networks and various problems arising in current and anticipated applica-tions over such networks. In this talk, optimization problems and algorithms fordistributed multi-agent networked systems will be discussed. The distributednature of the problem is reflected in agents having their own local (private)information while they have a common goal to optimize the sum of their ob-jectives through some limited information exchange. The inherent lack of acentral coordinator is compensated through the use of network to communicatecertain estimates and the use of appropriate local-aggregation schemes. Theoverall approach allows agents to achieve the desired optimization goal withoutsharing the explicit form of their locally known objective functions. However,the agents are willing to cooperate with each other locally to solve the problemby exchanging some estimates of relevant information. Distributed algorithmswill be discussed for synchronous and asynchronous implementations togetherwith their basic convergence properties.

The problem of interest is a convex constrained minimization of a networked-system problem, which has the following form:

minimizem∑

i=1

fi(x) subject to x ∈ X,

where m is the number of agents (nodes) in the network, each fi : Rn → R

is a convex function known only to agent i, and X ⊂ Rn is a closed convex

set. The agents are connected by a physical communication network allowingthem to share some information locally with their neighbors in the network.The algorithms for solving this system problem obey the network connectivitystructure and the privacy of agent’s objective functions. Both directed andundirected time-varying connectivity networks will be considered.

127

Universal gradient methods

Yurii Nesterov1

1 CORE/UCL, Louvain-la-Neuve, Belgium Yurii.Nesterov@uclouvain.be

In Convex Optimization, numerical schemes are always developed for somespecific problem classes. One of the most important characteristics of suchclasses is the level of smoothness of the objective function. Methods for nons-mooth functions are different from the methods for smooth ones.

However, very often the level of smoothness of the objective is difficult toestimate in advance. In this talk we present algorithms which adjust theirbehavior in accordance to the actual level of smoothness observed during theminimization process. Their only essential input parameter is the required ac-curacy of the solution. At the same time, for each particular problem classthey automatically ensure the best possible rate of convergence. We confirmour theoretical results by encouraging numerical experiments.

References

1. Yu.Nesterov Universal gradient methods, CORE Discussion Paper 2013/26(2013)

128

Regret in Optimal Stopping and It’s Relation toAmbiguity

Lazar Obradovic1

1 IMW, Bielefeld University, Bielefeld, Germany;lazar.obradovic@gmail.com

We focus on regret in optimal stopping in discrete time with finite horizon:an agent observes a stochastic process Xt, t = 1, . . . , n and considers regretprocess Rt = maxk=1...n f(Xk−Xt) which she tries to minimize ove all stoppingtimes. We are interested in identifying the optimal stopping time and comparingit with the solution of a related problem of maximizing the process Xt overstopping times. Difference of effects of ambiguity (in the sense intoroduced in[1]) on the two optimal stopping problems is analyzed using tools developed in([2], [3]).

In the simplest case where we take f to be identic mapping we show that,under very mild assumptions, the same stopping time optimizes original processand regret process. Before ambiguity is introduced we generalize several resultsfrom [3] which we use in further analyzis.

We give sufficient assumptions on the function f for the above results tohold before we turn to fundamental question: could the appropriate choice ofthe form of function f replicate the effects of ambiguity (in sense of optimalstopping times)? We give a surprising answer that deems the question void formost practical purposes, yet conclude with an interesting example where theanswer is affirmative.

The research is part of author’s PhD research at IMW, Bielefed, Germanyunder supervision of Professor Frank Riedel.

References

1. I. Gilboa, D. Schmeidler Maxmin Expected Utility with Non-Unique Prior,Journal of Mathematical Economics, 18, 141153, (1989).

2. T. Chudjakow, F. Riedel. “The Best Choice Problem under Ambiguity”IMWworking papers; 413. Bielefeld: Institute of Mathematical Economics, Biele-feld University, (2009).

3. F. Riedel. “Optimal Stopping with Multiple Priors”,Econometrica, Volume77, Issue 3, pages 857908, (May 2009).

129

Global climate stabilization controlled bystratospheric aerosols injections

Valeriy Parkhomenko

Dorodnicyn Computing Centre of RAS, Moscow, Russia; parhom@ccas.ru

Global warming climate changes are observed in recent decades. Thesechanges largely associated with anthropogenic increases in greenhouse gases inthe atmosphere (CO2 most important among them) [1]. The problem andopportunity of the global climate stabilization [2] at a current level are investi-gated.

The study is based on a three-dimensional hydrodynamic global climatecoupled model, including ocean model with real depths and continents config-uration, sea ice evolution model and energy and moisture balance atmospheremodel [3, 4].

The climate prediction calculations up to the year 2100, using CO2 growthscenario A2, proposed by IPCC are carried out on the first stage [4]. They givean increase in global mean annual surface air temperature at 2.2 degrees in theyear 2100.

Next, a series of calculations were carried out to assess the possibility of cli-mate stabilization at the level of the year 2010 by controlling emissions into thestratosphere of aerosol, reflecting part of the incoming solar radiation. Aerosolconcentration from the year 2010 up to 2100 is calculated as a controlling pa-rameter to stabilize mean year surface air temperature.

It is shown that by this way it is impossible to achieve the space and sea-sonal uniform approximation to the existing climate, although it is possiblesignificantly reduce the greenhouse warming effect.

Assumption of a uniform stratospheric aerosol space distribution can sta-bilize the mean atmosphere global temperature, but climate will be colder at0.1-0.2 degrees in the low and mid-latitudes and at high latitudes it will bewarmer at 0.2-1.2 degrees. In addition, these differences have strong seasonalmove they increase in the winter season.

The situation is slightly better when we allow latitude dependence of aerosolconcentrations. Compared to the previous case, the concentration is reducedin low and middle latitudes and is enlarged in high latitudes. However, theincrease in the aerosols concentration in the polar areas gives a weak effect,since there is little influence of solar radiation, while the greenhouse effect isstable.

It is assumed in the following numerical experiments that aerosol emissionsand concentration are zero from the year 2080. This leads to a rapid increase of

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the mean global atmospheric temperature, approaching the temperature with-out the aerosols to the year 2100.

The author was supported by the Russian Foundation for Basic Research(projects no. 11-01-00575, 11-07-00161) and Presidium RAS Basic ResearchProgram (no. 14).

References

1. Climate Change 2007 The physical Science Basis. Contribution of WorkingGroup 1 to the Fourth Assessment Report of IPCC . 2007, 989 p.

2. Mercer A.M., Keith D.W., Sharp J.D. Public understanding of solar radia-tion. Management Environ. Res. Lett. 2011. V. 6. P. 1-9.

3. Marsh R., Edwards N.R., Shepherd J.G. Development of a fast climate model(C-GOLDSTEIN) for Earth System Science.

4. V.P. Parkhomenko. Climate model with consideration World ocean deep cir-culation, Vestnik MGTU im. Baumana. Issue Mathematical Modelling, p.186-200 (2011) (in Russian).

131

The best ellipsoidal approximation of the attractiondomain in the path following problem for a wheeled

robot with constrained control resource

Alexander Pesterev1

1 Institute of Control Sciences RAS, Moscow, Russia; a.pesterev@javad.com

Stabilization of motion of a wheeled robot with constrained control resourcealong a given target path is considered. Robot’s motion is governed by thenonlinear affine system (kinematic, or simple-car, model)

xc = v cos θ, yc = v sin θ, θ = v tan δ/L, (1)

where xc, yc are coordinates of the target point located in the middle of the rearaxle, θ is the orientation angle, v ≡ v(t) is the vehicle velocity, L is the distancebetween the front and rear wheels, δ is the turning angle of the front wheels,and the dot over the symbol denotes differentiation with respect to time. Owingto the equation u = tan δ/L relating the angle δ with the curvature u of thetrajectory described by the target point, we may consider u as the control. Sincethe turning angle δ is bounded by δ < π/2, the control is also bounded: |u| ≤ u.The target path is given parametrically by a pair of functions X(s), Y (s), wheres is the arc length.

The linearizing feedback for system (1) in the path following problem isfound to be [1]

u = −σ(z) cos3 ψ +k cosψ

1− kd, (2)

where d is the deviation of the target point from the target path, ψ is theangular deviation (the angle between the velocity vector and the tangent to thetarget curve at the point closest to the robot), k ≡ k(s) is the path curvature,z ≡ [z1, z2]

T = [d, tanψ]T, σ(z) = λ2z1 + 2λz2, λ > 0. If the control resourceis unbounded (the front wheels can turn through an angle of π/2), feedback(2) guarantees asymptotic decrease of deviation. In the case of the constrainedcontrol resource, stabilization is not guaranteed for arbitrary initial conditions,and we arrive at the problem of finding an estimate of the attraction domain.We look for an ellipsoidal approximation of the desired estimate in the formΩ = z : zTPz ≤ α2 and put the problem of finding an ellipse of the maximumpossible area, which is further referred to as the best ellipsoidal approximationof the attraction domain.

Finding an ellipsoidal estimate of the attraction domain was shown to reduce[2] to joint solving an LMI system and a scalar inequality. In this paper, weshow that the problem of finding the best ellipsoidal approximation reduces

132

to solving a standard constrained optimization problem for a function of twovariables, as stated in the theorem below. Let us introduce the following threefunctions: U0(α) = u − k/(1 − kα), where k is the maximum curvature of thetarget path, σ0(q1, q2) =

√1− 2q1 + 4q2, and

β0min(q1, q2) =

2q1q2+2q2−q21−√q1(q1−4q2)(q

21−4q2)

2(1−q1)2 , q1 6= 1,q22

4q2−1, q1 = 1.

Theorem. Let α∗(q1, q2) be solution of the scalar equation

U0(α) = αλ2β0min(q1, q2)σ0(q1, q2)

and let q∗1 , q∗2 be solution of the problem of maximization of the function

D(q1, q2) = λ2α4∗(q1, q2)

(q2 − q21

4

)

under the constraint (q2 − q1 − 1)2 + (q1 − 2)2 ≤ 4. Then, matrix P of the bestellipsoidal approximation of the attraction domain is given by

P =P

det P, P = α2

∗

(λ2q∗2 λq∗1/2λq∗1/2 1

).

This work was supported by the Department of Power Engineering, Me-chanical Engineering, Mechanics, and Control Process of Russian Academy ofSciences (Program no. 1 “Scientific foundations of robotics and mechatronics”).

References

1. A.V. Pesterev. “Synthesis of a stabilizing control for a wheeled robot followinga curvilinear path,” Automation and Remote Control, 73, No. 7, 1134–1144(2012).

2. A.V. Pesterev, L.B. Rapoport. “Construction of invariant ellipsoids in the sta-bilization problem for a wheeled robot following a curvilinear path, ” Journal,Automation and Remote Control, 70, No. 2, 219–232 (2009).

133

Control of oscillations in essentially nonlineardynamic systems with multiple degrees of freedom

Lev F. Petrov1

1 Plekhanov Russian University of Economics, National Research UniversityHigher School of Economics, Moscow, Russia; LFP@mail.ru

We consider the essentially non-linear dynamic systems with multiple de-grees of freedom, whose behavior is described by a system of ordinary differentialequations of the form

xj(t) + j2(k2j2 +D1)xj(t) + xj(t)j2

4(

N∑

m=1

m2x2m(t)) + δ1xj(t) = Qj(t), (1)

j = 1, 2, ..., N ;Qj(t) = Qj(t+ T )

Note that for some forms of oscillations the sign in front linear term xj(t) canbe negative. This corresponds to a qualitative change in the type of subsystem.Thus the individual subsystems can have a negative stiffness.

In deterministic systems of type (1) even in the case of one degree of freedom(N = 1) in the different ranges of values of parameters of a periodic solutionmay have a variety of shapes. Among them we note the periodic solutionsof simple form, complex polyharmonic periodic solutions, chaotic behavior [1].The interaction of different types of oscillations is possible in a nonlinear systemwith one degree of freedom, for example, the combination of forced, parametricand self-excited oscillations.

Even more complicated behavior of solutions of the system the type (1)states, taking into account multiple degrees of freedom (N > 1). In this inter-action occur the oscillations that correspond various degrees of freedom. Thisinteraction is the source of new dynamic regimes. Even in a linear dynamicalsystem with two degrees of freedom is known effect of dynamic damping. Innonlinear dynamical system interacting vibrations corresponding to the differ-ent degrees of freedom generates various effects of amplification, damping andthe appearance a new complex polyharmonic periodic solutions, changes in therange of parameters where detected chaotic regime.Thus there is a change inthe field of manifestation a strange attractor, which took place in the corre-sponding system with one degree of freedom, due to the impact of fluctuationson the other degrees of freedom.

Even more varied interaction effects of different types of oscillations instrongly nonlinear dynamic systems with several degrees of freedom can be ex-

134

pected, taking into account the oscillations of different nature (forced, paramet-ric, self-excited oscillations), corresponding to the individual degrees of freedom[2].

Considering the possibility of increasing the dimension of the dynamic sys-tem of type (1) can be used for targeted control of dynamic regimes correspond-ing to one of the forms of oscillations, similar to how this is realized in linearsystems using dynamic damping.

Note that we use a technique of analysis [3] of dynamical systems of type(1) allows one to study these systems more general form, in particular - witha nonlinear dissipation. It is possible to take into account different dissipa-tive characteristics of the oscillations corresponding to the different degrees offreedom. In this case, there are many possible manifestations of the effect ofchanging the dissipative characteristics manifested by the interaction of oscilla-tions on individual degrees of freedom. A special interest may cause the studyof nonlinear dissipation on the manifestation of deterministic chaos.

It is also possible not analytical, and table expression dependencies includedto the studied dynamical system. There is no need to approximate the tabulardata dependence analysis, it is only necessary to use the appropriate interpola-tion accuracy. This allows to include to the studied model of nonlinear dynamicsystem real relationships derived from experimental data.

References

1. P.J. Holmes “A nonlinear oscillator with a strange attractor”, PhilosophicalTransaction of the Royal Society, London, 292, pp. 419-448.

2. L.F. Petrov “Nonlinear effects in economic dynamic models”. NonlinearAnalysis 71 (2009) e2366 - e2371.

3. L.F. Petrov “Interactive optimization as a tool for finding the complex peri-odic solutions in nonlinear dynamics”. Proceedings of III International con-ference on optimization methods and applications (OPTIMA-2012) p.p. 213-218.

135

On price competition in Stackelberg location-pricemodel

Alexander Plyasunov1

1 Sobolev Institute of mathematics, Novosibirsk, Russia; apljas@math.nsc.ru

We study a new model of competitive location and pricing, which can be de-scribed with the help of the next Stackelberg type leader–follower game. Givenfinite set of facility locations and the finite number of markets. The playerscompete for demand for a homogeneous product in spatially separated markets.Demand in each market is price-sensitive and is given by a known decreasingfunction of price. The competing players use delivered pricing, i.e. the sellercharges a specific price in each market area, which includes the freight cost, andtakes care of transport. We assume that the client demand is concentrated ata finite set of discrete points.

In the first stage of the game the leader opens his own p facilities. At thesecond stage the follower opens his own r facilities. The fixed facilities ownedby some player will be compete with each other, as well as with any facilityowned by its competitor. Once the locations of the facilities are fixed the leaderand the follower will compete on price. We assume that each facility will notprice below its marginal cost in each market. Customers always buy from thefacility that offers the lowest price in the market they belong to. If more thanone facility offer the same lowest price, customers buy from the facility withthe minimum marginal delivered cost. This is justified by the fact of a facilitywith less marginal cost then its competitor can offer a lower price and servethe market. So the leader will play a Bertrand game with the follower in eachmarket in order to maximaze its profit. Price competition leads each competingfacility to monopolize a group of markets, which are the ones for which themarginal delivered cost of the facility is less than the minimum delivered costof its competitors. Then each player sets the optimal price in each one of itsmonopolized markets. Each player tries to maximize his own market share.The goal of the game is to find p points for the leader facilities to maximize hismarket share.

This model is able to capture some of the principal trade-offs involved infacility location, including demand cannibalization, market expansion. In par-ticular, we can study the issues associated with the elasticity of demand.

The model is formulated as a bilevel linear Boolean programming problem.Assume that the profit in any market is determined by some monotonicallydecreasing function of the minimum delivered cost of a monopolist. Then we

136

have the following statement.

Theorem 1. The problem belongs to the class Poly-APX and is ΣP2 -hard.For this problem we develop an alternating heuristic, based on the exact ap-

proach for the follower problem. The idea of alternating methods is well-known[1]. Given a solution for the leader, the best-possible solution for the followeris computed. Once that is done, the leader may tentatively assume the roleof the follower and reoptimize his set of facilities by solving the correspondingproblem for the given solution of the follower. This process is then repeateduntil a termination condition is satisfied. Scheme of this algorithm is as follows:

1. Build initial solution. Set best goal function (record) value to zero.Main cycle: for a given number of iterations do following:2. Solve follower problem for current leader placement. If new goal functionvalue is grater then record, then update record.3. Put leaders facilities exactly in the same points as follower did on previousturn. Go to step 2.

Best found solution in the end gives as answer. As initial solution we chooserandom set of p points. On the second step of this scheme exact solution offollower problem is needed. Calculations are terminated after a sufficientlylarge number of iterations. Computational results are discussed.

The author were supported by the Russian Foundation for Basic Research(projects no. 13-07-00016, 13-06-00023).

References

1. E. Carrizosa, I. Davydov, Yu. Kochetov. A new alternating heuristic for the(r—p)-centroid problem on the plane. Operations Research Proceedings 2011,Springer, (2012), P. 275-280.

137

Least Square Method and Improper MP Problems

Leonid Popov

Institute of mathematics and mechanics UB RAS, Ekaterinburg, Russiapopld@imm.uran.ru

In the report a new modification of the well-known least square method isdiscussed. The modified method may be applied not only to usual, solvablemathematical programming (MP) problems, but also to improper (ill-posed,contradictory) problems of the 1st kind. If an original problem is solvable thenthe modified method finds its usual solution. Otherwise, some generalized so-lution having very natural substantial interpretation is automatically provided.

Let us consider the problem

inff0(x) : fj(x) ≤ 0 (j ∈ 1,m), x ∈ Rn, (1)

where all the functions f0(x), f1(x), . . . , fm(x) defined on Rn are convex. As-sume the problem (1) may be infeasible (improper of the 1st kind [1]) and define

u = arg inf ||u|| : u = [u1, . . . , um] ∈ clΩ,

where Ω = u : M(u) 6= ∅, M(u) = x ∈ Rn : fj(x) ≤ uj (j ∈ 1,m). Using uone can generate the problem with improved right hand sides

inff0(x) : fj(x) ≤ uj (j ∈ 1, m), x ∈ Rn. (2)

Argument of this problem (if exists) may be considered as some sort of gener-alized solution of the improper problem (1).

Note, that for the proper problems u = 0 and the generalized solutionsintroduced above coincide with the usual ones.

To find generalized solution let us write the standard least square function

Φ(x, v) = ([f0(x)− v]+)2 + Φ0(x), Φ0(x) =∑m

i=1[f+i (x)]2,

where w+ = max0, w for a scalar w, parameter v estimates the optimal valuev of the improved problem (2) from below.

The modified least square method consists of iterations of the form (fororiginal method see [2]):

v0 < v, xk ∈ Argminx

Φ(x, vk), vk+1 = f0(xk), k = 0, 1, . . . . (3)

The convergence properties of the modified method are as follows.

138

Theorem 1. Let u ∈ Ω and M(u) be bounded. Then the process (3)generates the sequence of pairs (vk, xk) with the properties:

a) f0(xk) ր v;

b) Φ0(xk) ց minx

Φ0(x) (= |u|2);c) xk → Arg (2).

The modified method has a simple geometric interpretation. Let

K=(u, v) : v ≥ v(u), u ∈ Ω, v ∈ R, L=(0, v) : 0 ∈ Rm, v ∈ R.

Under the assumptions above K is nonempty convex closed set. The set L is aline in Rm+1. In a proper case it is well-known that K ∩ L 6= ∅ and the pair(0, v) is the common point of K and L having minimal v-component. In animproper case K ∩ L = ∅. It is easy to see that in fact the relations (3) realizethe cyclic projection of pairs (u, v) onto sets K and L; at first onto K, thenonto L and onto K again, etc.

Note that there are many publications devoted to analysis of cyclic projec-tion methods onto a finite series of closed convex sets both in the case whenthese sets have a common point as well as in the case when such point does notexist [1, 3].

The author was supported by the Russian Foundation for Basic Research(projects no. 13-01-00210 and no. 13-07-00181).

References

1. I.I. Eremin, Vl.D. Mazurov, N.N. Astafiev. Improper problems of linear andconvex programming, Nauka, Moscow (1983). (in Russian)

2. V.Yu. Lebedev. “Convergence of the weighted functional method in convexprogramming problems,” USSR Computational Mathematics and Mathemat-ical Physics, 17, 3, 765–768, (1977).

3. I.I.Eremin, L.D. Popov. “Closed Fejer cycles for incompatible systems of con-vex inequalities,” Russian Mathematics (Izvestiya VUZ. Matematika), 52, 1,8–16. (2008).

139

On the decomposition of the Ramsey model

I.G. Pospelov

Dorodnicyn Computing Centre of RAS, Moscow, Russia; pospeli@yandex.ru

Perhaps the most important achievement of the mathematical economics foralmost 200 years of its existence is the first theorem of welfare theory, which inthis report can be summarized as:

The problem of optimal planning of the economy in the interests of theconsumer at technological constraints can be formally transformed into a non-cooperative game of aggregate economic agents that are naturally interpreted asconsumer, producer, marketer, and (in dynamic problem) bank. From economicpoint of view, this game is a description of the competitive equilibrium.

Model of the economy are usually built in the opposite direction, startingfrom the equilibrium interaction of certain agents. But those sophisticatedmodels of economic dynamics, which I described at the previous conferenceswere built as a decomposition of the planning problem ”spoiled” by observed inreality institutional constraints.

This report discusses the process of decomposition on the example of asimple model of economic growth - the Solow-Ramsey one. It will be shownhow to explain the emergence of many economic mechanisms and institutionsrequirements as decentralized implementation of the optimal plan.

References

1. I.G. Pospelov.Models of economic dynamics, based on the balance of forecastsof economic agents, Computing Centre, Moscow (2003). [In Russian]

2. A.A. Petrov, I.G. Pospelov. “Mathematical models of the Russian economy,”Herald of the Russian academy of sciences, 79, I. 3, 205–216 (2009).

1. N.P. Pilnik, I.G. Pospelov. Description of the objectives of the firm in adynamic general equilibrium model, Computing Centre, Moscow (2009). [InRussian]

140

Edge multicoloring of weighted unicyclic graphs

Artem Pyatkin1

1 Sobolev Institute of Mathematics, Novosibirsk, Russia; artem@math.nsc.ru

Let a positive integer w(e) (the weight of e) is assigned for every edge e of thegraph G = (V,E). Denote by ∆(G) the weighted degree of the graph G equalto the maximum sum of the weights of edges incident to a same vertex. Let ussay that an interval I = [a, b] is normal if a and b are non-negative integers andb ≥ a. Then a is the left and b is the right end of the interval I = [a, b]. Denotethem by l(I) and r(I) respectively. The length of the interval [a, b] equals tob − a. The edge multicoloring of the graph G is an assignment of a normalinterval to each edge such that its length is at least the weight of the edge; suchinterval is called a multicolor of this edge. We say that a multicoloring is properif the multicolors of adjacent edges do not intersect by the inner points. Themaximum of the difference r(I) − l(J) among all multicolors I and J used atthe multicoloring is called the number of colors of these multicoloring. The edgemultichromatic number µ′(G) is the minimum number of colors necessary for aproper multicoloring of all edges of the graph G.

The problem of finding the edge multichromatic number generalizes twowell-known NP-hard problems: the edge coloring of a graph [2] (the case whenall weights are equal to 1) and the scheduling problem open shop without pre-emptions [1] (the case when the graph G is bipartite). Note that the moregeneral problem of the vertex multicoloring of a graph was considered in [4].

For the edge chromatic number the Shannon’s bound [3] is known: χ′ ≤⌊3∆/2⌋. In the scheduling theory there is a ”folklore” conjecture [5] that theminimum length of the schedule in the open shop problem without preemptionsis at most one and a half times the trivial lower bound. The following conjecturegeneralizes these two statements:

Conjecture 1. For every weighted graph G = (V,E) the bound µ′(G) ≤⌊3∆/2⌋ holds.

Note that this bound is trivially attainable: just consider a triangle with thesame weights of the edges. However, it can be attained even on trees. To showit, consider a subdivision of the star K1,2t+1 and let the weights of the edgesadjoining to the center of the star be 1, and the weights of the remaining edgesbe 2t. It is easy to check that in this case ∆ = 2t+1 and µ′ = ⌊3∆/2⌋ = 3t+1.

In this paper this conjecture is proved for unicyclic graphs, i. e. the con-nected graphs having exactly one cycle. The following theorem is proved.

Theorem 1. If a connected graph G satisfies the formula |V (G)| = |E(G)|then µ′(G) ≤ ⌊3∆(G)/2⌋.

141

The proof of the theorem is based on the following ”interval packing exten-sion” lemma. Let I = [0, ⌊3∆/2⌋]. We say that the intervals I1, I2, . . . , Ik arepacked correctly into the interval I if they do not intersect by the inner pointsand none of Ij lies in the interval [⌊∆/2⌋+1,∆−1] (i. e. for every j = 1, 2, . . . , kwe have l(Ij) ≤ ⌊∆/2⌋ or r(Ij) ≥ ∆).

Lemma 1. Let w1, w2, . . . , wk be non negative integers whose sum is atmost ∆; each wj is the length of some interval Ij , j = 1, 2, . . . , k. Let theintervals I1, I2, I3 are correctly packed into the interval I = [0, ⌊3∆/2⌋] suchthat l(I1) = 0 and r(I2) = ⌊3∆/2⌋. Then the intervals I4, I5, . . . , Ik can bepacked correctly into the interval I so that none of them intersect the intervalsI1, I2, I3 by inner points.

Remark. Lemma 1 remains true even if some of the intervals I1, I2, I3 havezero length.

The work was supported by the Russian Foundation for Basic Research(projects no. 12-01-33028, 12-01-00090, 12-01-00093, 12-01-00184, and 13-07-00070) and by the grants of the target program of SB RAS (integration projects7B and 21A).

References

1. M.R. Garey, D.S. Johnson Computers and Intractability: A Guide to theTheory of NP-Completeness. San Francisco: Freeman, 1979. 314 P.

2. I. Holyer The NP-completeness of edge-coloring // SIAM J. Comput. 1981.V. 10. 4. P. 718-720.

3. C. E. Shannon. A theorem on coloring the lines of a networks // J. Math.Phys., 1949. V. 28, P. 148–151.

4. V.G. Vizing.On vertices multicoloring of weighted graphs // Discrete analysisand operations research, Ser. 1, 2007. V. 14, 4. P. 17–26. (In Russian).

5. G.J. Woeginger. Open problems in the theory of scheduling // Bulletin of theEATCS, 2002. 76. P. 67–83.

142

Freight Car Routing Problem

Ruslan Sadykov1, Alexander Lazarev2, Alexey Karpychev3

1 INRIA Bordeaux Sud-Ouest, Talence, France; ruslan.sadykov@inria.fr2 Institute of Control Sciences RAS, Moscow, Russia;

M.V.Lomonosov Moscow State University;National Research University Higher School of Economics; jobmath@mail.ru

3 Institute of Control Sciences RAS, Moscow, Russia; karpich@frtk.ru

At present time in most countries for freight transportation interaction ofseveral companies is necessary. First company is freight, it has got railwaycars and directly executes customer orders for the transportation of good byproviding cars. Second company is railway, it provides transportation of thecars under special rules. We consider the freight car routing problem.

We know initial car distribution, orders, profits for delivering and tariffs forstandings and for empty transfers. Most orders we can accept or reject. Weshould find subset of profitable orders and optimal logistic for empty cars. Wehave sets of stations, car models, good types, orders and sources (station can besource of cars). For all stations, car models and good types we know 2 functions:price and duration of empty transfer. Price depends on the last good type. Wehave also planning horizon.

For the formulation of integer programming problem we’ll use integer vari-ables for empty transfers, standings and order transfers, binary variable fororders (that indicates delivered or not). Problem can be formulated as problemof finding several flows with maximal cost in oriented graph without cycles andadditional constraints. Every flow corresponds to movement the cars of thesame model. Objective function maximizes profit, that equals difference be-tween profit for delivered orders and charges for standings and empty transfers:

max∑

pq,t−rqzwqt −

∑c · yαcitw −

∑M(i, j, k, w)xc

′c′′kijtw (1)

Constraints (2) - (3) give minimal and maximal number of cars for delivering

orders:

∑

w∈Wq

rq+∆q∑

t=rq

zwqt ≤ vmaxq uq (2)

143

∑

w∈Wq

rq+∆q∑

t=rq

zwqt ≥ vminq uq (3)

Constraints (4) - (5) are balance equations for cars transferred for the order

and freed after order:

y1citw = y1ci,t−1,w +∑

xc′ckj,i,t−D(j,i,k,w),w −

∑zwqt, (4)

y2ckitw = y2cki,t−1,w −∑

xcc′k

ijtw +∑

zwq,t−dq +∑

~vs, (5)

Then we reformulate the problem in terms of routes. We use integer vari-ables only for numbers of cars in routes. Then we solve its linear relaxation bycolumn generation.

This approach gave good results for tests from freight company. In severalcases working time was similar and in some cases it was significantly reduced.

The authors were supported by the Russian Foundation for Basic Research(project no. 11-08-13121).

References

1. L.A. Wolsey. Integer Programming, New York (1998).

2. A.A. Lazarev, E.G. Musatova, E.R. Gafarov, A.G. Kvaratskheliya. Theoryof Scheduling. The tasks of transport systems management. - M: PhysicsDepartment of M.V.Lomonosov Moscow State University, 2012 – p.160

144

Controllability and optimality of one-dimensionalmovement

Vladimir Savelyev

University of Nizhni Novgorod, Nizhni Novgorod, Russia;vpsavelyev@rambler.ru

Some proprieties of controllability and optimal trajectories of the locallycontrollable system

x+ f(x, x) = u(t), q ≤ u(t) ≤ p, (1)

are considered. The autonomous systems, corresponding the values p(q) will benamed p-system (q-system). They are supposed to satisfy some rather naturalconditions of “effective braking”. To describe limits of a controllability set weintroduce so-called generalized trajectories of these two systems, using the α-separatrices and ω-separatrices of their saddle-points, and another two systems:pq-system coincides with the p-system in the half-plan G+ = (x, x) ∈ R2 | x ≥0 and with the q-system in the half-plan G− = (x, x) ∈ R2 | x ≤ 0; qp-systemcoincides with the q-system in the half-plan G+ and with the p-system in thehalf-plan G−. The simplest kind of a limit of a controllability set is formedby a close generalized trajectory of qp-system having the point (0, 0) inside.An uncontrollability set N usually consists of some quantity of connected setsthat we call as “connected components of N”. The simplest kind of a limitof a connected component of N is formed by a close generalized trajectory ofpq-system having the point (0, 0) outside.

We prove that the limit of every connected component of N , except twotypes mentioned above, contains at least one saddle-point of p-system or q-system with its ω-separatrices (so-called “generating saddle-point”).

The structure of optimal trajectories minimizing the linear combination T +αE (α ≥ 0) of the time and the consumed energy to transfer the system (1)from an initial state to a final state is studied.

References

1. V.P.Savelyev. “On Classification of Connected Components of an Uncontrol-lability Set in a Nonlinear Oscillator,” Vestnik of Lobachevsky State Univer-sity of Nizhni Novgorod, No. 3, Part 1, 155–162 (2012).

145

Object and morphism symmetry of optimalityconditions

Simon Serovajsky1

Al-Farabi Kazakh National University, Almaty, Kazakhstanserovajskys@mail.ru

Consider an abstract optimization control problem. Let us have a contin-uously operator A : U × Y → Z and a functional I : U × Y → R, whereU, Y, Z are Banach spaces. The state y = Lu are determined uniquely by theequality A(u, y) = 0 for all control u ∈ U . The pair w = (u, y) is called theobject of the optimization control problem, and the pair F = (I,A) is called itsmorphism. We have the minimization problem of the functional J , determinedby the equality J(u) = I(u,Lu) on the space U .

The necessary optimality conditions can be obtained from the stationarycondition J ′(u) = 0, using the Implicit Function Theorem. Suppose the deriva-tive Ay(w) of the mapping y → A(u, y) at the point y = Lu is invertible. Thenwe find the derivative L′(u) = −

[Ay(w)]

−1Au(w), where Au(w) is the deriva-tive of the mapping u→ A(u, y) at the point u. Using the Composite FunctionTheorem, we get

J ′(u) = Iu(w) −[Au(w)

]∗[Ay(w)

]∗−1

Iy(w).

Since we have the equality

Iu(w) −[Au(w)

]∗p = 0,

where p is the solution of the adjoint equation

[Ay(w)

]∗p = Iy(w).

We could determine another form of the optimality conditions with highlevel of the symmetry, which is not use the adjoint state p. Differentiate thefunctional J and the operator B, determined by the formula Bu = A(u,Lu).Using the stationary condition we obtain the equalities

Iu(w) + Iy(w)L′(u) = 0, (1)

Au(w) + Ay(w)L′(u) = 0. (2)

146

These relations have a morphism symmetry. If we rearrange the state operatorand the cost functional, we get the same equalities.

Besides, the state equation is uniquely solvable. So the sets of the controlsand the states are isomorphic. Therefore we can interpreted the state equationas a means of the finding of the control u = My by the given state y, whereM = L−1. Then we obtain the minimization problem for the functional K,determined by the equality Ky = I(My, y) on the space Y . Determine itsderivative. Using stationary condition we have the equalities

Iy(w) + Iu(w)M′(y) = 0, (3)

Ay(w) + Au(w)M′(y) = 0. (4)

These relations have the morphism symmetry too. However we have also theobject symmetry because we get the system (1), (2) after rearranging of thestate and the control. Take into account here that the derivatives L′(u) andM ′(y) are mutually invertible by the Inverse Function Theorem.

These results can be denoted uniformly with using of Jacobian of the mor-phism F at the object w. It is the matrix operator with elements Iu(w), Iy(w),Au(w), Ay(w). Determine the colomn vector λ(u) with components E, L′(u)and the colomn vector µ(y) with components M ′(y), E, where E is the unitoperator. Then the system (1) – (4) can be transformed to the equalities

F ′(w)λ(u) = 0; F ′(w)µ(y) = 0.

The finite dimensional cases of these results are considered as examples.Particularly the necessary extremum condition for the minimization of the mul-tivariable function with much equality constraints are the equality of the corre-sponding Jacobian to zero. By the morphism symmetry we prove that it is theextremum condition for the minimization of the arbitrary function of the givenfamily, where other functions are used for the constraints. These ideas are usedalso for an optimization problem with state constraint. Using object symmetrywe can apply the state variation for obtaining necessary optimality conditions.

147

Comparing of approximation of functional gradientand gradient of functional approximation for heat

conduction optimization problem

Ilyas Shakenov

Al-Farabi Kazakh National University, Almaty, Kazakhstanilias.shakenov@gmail.com

In the work we consider inverse boundary problem in one dimensional heatconduction equation.

ut = uxx + f(x, t), 0 < x < L, 0 < t < T, (1)

u|t=0 = ϕ(x), (2)

ux|x=0 = b(t), (3)

ux|x=L = y(t), (4)

Right boundary function y(t) is unknown. To determine this value we can

use an additional information u(0, t) = a(t). We convert this problem to op-timization, in which we are required to minimize the functional on (1) – (4):

I(y) =

T∫

0

(u(0, t; y)− a(t)

)2dt→ min

If we find control y(t) which gives a zero value to the functional then the

additional information is fulfilled.We use one familiar method to solve the problem by constructing an iterative

process.

yn+1(t) = yn(t)− αnI′(yn(t)

),

where α > 0. The first problem is to determine what is I ′ and how to find it?

Using an apparatus of Gateaux derivative we derive the theorem: TheoremThe Gateaux derivative of functional I in point y(t) is equal to ψ

(L, t), where

ψ(x, t)is a solution of adjoint problem

ψt + ψxx = 0, (5)

ψ(x, T

), (6)

148

ψx(L, t), (7)

ψx(0, t) = −2(u(0, t; y)− a(t)

), (8)

If we write an approximation of direct problem (1) – (4), functional I , adjoint

problem (5) – (8) and solve it, we meet a problem that there is an unavoidableerror on right-side of interval 0 < t < T while the left side gives very goodsolution. We can get rid of the problem by cutting the right side and taking thesolution only on the part of the interval. But we consider one more way. Basedon approximation of (1) – (4) and functional I we derive an adjoint problemin discrete form and, consequently, the gradient of an approximation also indiscrete form. After comparing with approximation of the gradient we concludethat they are different. What follows is a comparing of two methods.

The only difference we meet in two discrete problems is a form for the

boundary condition, namely, approximation of (8) can be written asψ

j1−ψ

j0

h=

−2(uj0 − aj

), j = M, . . . , 0. While gradient of approximated problem gives us

a slightly different result,ψ

j1−ψ

j0

h= −2

(uj+10 − aj+1

), j = M, . . . , 0. (Here M

denotes a number of splits of time interval).Apparently, it is much more difficult to derive a gradient of approximation

of functional. That is why so far usually one found a gradient in continuousform and then found approximation of that. In the work, we perform a series ofexperiments and check whether the gradient of approximation allows us to avoida problem of unremovable errors in the left side of time interval. The results wederived show that the problem does not go out and there is a significant errorson the right side of interval 0 < x < L.

So we make a conclusion that we can work with approximation of continuousform of gradient and do not make a lot of transformations to obtain a gradientof approximated problem.

149

The recurrent estimating of unknown parameter ofPDF using control parameter

Kanat Shakenov

Al-Farabi Kazakh National University, Almaty, Kazakhstanshakenov2000@mail.ru

In the given work we construct asymptotically normal and asymptotically ef-ficient estimate of parameter x0 in multivariable estimating problem of unknownparameter x if there is an additional control parameter z, which can chosen bya statistician. Basically, this problem can be led to an optimization problemof a process that we observe with an error, if a priori distribution of unknownparameter is given. But in this work we introduce a method that is based onrecurrent estimating of parameter x and that value of z, for which the observa-tion is the most profitable. Let f(y|x, z) is distribution density with respect tomeasure ν(dy) that depends on parameters x and z, x ∈ X ⊂ El, z ∈ Z ⊂ Ek.Parameter x = x0 is unknown and must be estimated by observations withdensity f . Parameter z has a meaning of control and can be chosen by a statis-tician. Problems consists in selecting of control (plan) Zi ∈ Z, i = 1, 2, . . . that

provides the most quality of estimating in sense of minimal value of E(xn−x0

)2.

Asymptotically normal recurrent control. Let consider the procedure:

Xn+1 −Xn =I−1

(Xn, Zn+1

)f ′x

(Yn+1|Zn+1, Xn

)

(n+ 1) f(Yn+1|Zn+1, Xn

) , Zn+1 = z(Xn). (1)

It is also can be shown that function

M(x, z) =

∫ln

(f(y|x, z)f(y|x0, z)

)f(y|x0, z)ν(dy) (2)

has a maximum point when x = x0 for any value of z. Assume: 1. Func-

tions f(y|x, z), M(x, z) are twice differentiable by x and equality (2) and∫f(y|x, z)ν(dy) can be twice differentiated under the integral sign with re-

spect to x. 2. For any ε > 0, z ∈ E1 supε<|x−x0|<ε−1

m(x, z)(x − x0) < 0, where

m(x, z) = ∂M(x,z)∂x

and function ∂m(x,z)∂x

is continuous. 3. Function I(x, z) iscontinuous positive and for each x it has unique maximum Imax(x) with respect

150

to z and Imax(x) = I(x, z(x)

), function z(x) is continuous. 4. Function

G(x) = I−2(x, z(x)

) ∫(f ′x(y|x, z)f(y|x, z)

∣∣∣z=z(x)

)2

f(y|x0, z(x)

)ν(dy)

is continuous and increase no fast than quadratic if |x| → ∞. 5. For some ε > 0

sup|x−x0|<ε

∫

∣∣∣ f′x(y|x, z)

f(y|x, z)

∣∣z=z(x)

∣∣∣>R→∞

(f ′x(y|x, z)f(y|x, z)

∣∣∣z=z(x)

)2

f(y|x0, z(x)

)ν(dy) → 0,

that is sup of the indicated integral uniformly convergent in some neighborhoodof point x0.

Theorem If conditions 1 – 5 are true then procedure (1) gives consistentand asymptotically effective estimate of x0 in strong sense and asymptoticallyoptimal control (optimal plan of the experiment).

Look through: [1], [2], [3], [4], [5], [6].

References

1. Kanat Shakenov. ”The Solution of the Inverse Problem of Stochastic OptimalControl,” Rev. Bull. Cal. Math. Soc., 20, No. 1, 43–50 (2012).

2. M.B. Nevelson, R.Z. Hasminsky. The stochastic approximations and recur-rent estimating, Nauka, Moscow (1972).

3. D. Blackwell, M.A. Girshick. The theory gambles of stochastic solutions, IL,Moscow (1958).

4. A.A. Feldbaum. The foundational theory of optimal automatics system,Nauka, Moscow (1966).

5. I.A. Ibragimov, R.Z. Hasminsky. ”The asymptotic conducts of certain sta-tistical estimations,” The Probability Theory and Applications, 17, No. 3,(1972).

6. H. Robbins, S. Monro. ”A stochastic approximation method,” Ann. Math.Statist., 22, No. 1, 400–407 (1951).

151

Triangulations of polyhedra and convex conesand realization of their f-vectors

Valery Shevchenko

University of Nizhni Novgorod, Nizhni Novgorod, Russia; shev@uic.unn.ru

Consider the set P of solutions to a system of linear inequalities over thefield of rationals Q

ai0 +

d∑

k=1

aikxk ≥ 0, i = 1, . . . ,m, (1)

that is called a polyhedron and for l = 1, . . . , m the convex cone A∗l

x0 ≥ 0; ai0x0 +d∑

k=1

aikxk ≥ 0, i = 1, . . . , l. (2)

Let bj be the j-th column of the matrix B, B(J) is its submatrix (bj , j ∈ J),B∠ is the set By, y ≥ 0 of all nonnegative combinations of columns in B.Denote by Bl the integer (d+1)×nl matrix such that A∗

l = B∠l , with nl being

minimal possible number of the columns.For l = 1, . . . ,m a triangulation of the cone A∗

l with nodes from a set B = Blis the set Tl = S1, . . . , St of Sτ such that the following conditions hold:

1) Sτ ⊆ 1, . . . , n,2) |Sτ | = r = rankB(Sτ ) = rankB,

3) B∠ =t⋃

τ=1

B∠(Sτ ),

4) B∠(Sτ ) ∩B∠(Sσ) = B∠(Sτ ∩ Sσ).For k = 0, . . . , d denote by Γk,l(Sτ ) the set of all k-dimensional faces of the

cone B∠(Sτ ). Denote by ∆k,l =t⋃

τ=1

Γk,l(Sτ ) the set of all k-dimensional faces of

the simplicial complex ∆l =d⋃k=0

∆k,l. Let fk,l = |∆k,l|, fl = (f0,l(∆), . . . , fd,l(∆))

and fl(λ) =d∑k=0

fk,lλk.

It is convinient to write the polynomial fl(λ) using another basis in the form

fl(λ) =

d∑

k=0

γkλk(1 + λ)d−k = γl(λ). (3)

152

An integer sequance γ = (γ0, γ1, . . . γd, 0, . . . ) is called d-realized, if thereexist matrices Al, Bl and a triangulation Tl of the cone B∠, such that thecondition (3) holds.

For any natural numbers a and i there exists a unique binomail i-expansionof a =

(aii

)+(ai−1i−1

)+· · ·+

(ajj

), where ai > ai−1 > · · · > aj ≥ j ≥ 1. The number

a<i> =(1+ai1+i

)+ · · ·+

(1+aj1+j

)is called the i-th pseudopower of a; 0<i> = 0.

Theorem 1.1 [1]. If there exists k, such that γk+1 > γ<k>k , then γ is not realized for any

d.2. If γk+1 ≤ γ<k>k , for k = 1, . . . , d− 1, then γ is (2d)-realized.

The following theorem allows us to find minimal d, such that the sequenceγ is d-realized.

Theorem 2. The integer sequence γ = (γ0, γ1, . . . ) is d-realized if and onlyif the following conditions hold:

1. γ0 = 1, γi ≥ 0 for i = 1, . . . , d and γk = 0 for integer k ≥ d,2. γi ≥ γd−i ≤ γd−i−1 for i = 1, . . . , ⌊ d

2⌋,

3. γi+1 − γj−i ≤ (γi − γj+1−i)<i> for j = d, . . . , 2d and i = 1, . . . , ⌊ j

2⌋.

Corollaries from this result are discussed. Missing information can be foundin the cited works.

References

1. F.S. Macaulay. “Some Properties of Enumeration in the Theory of ModularSystems,” Proc. London Math. Soc., 26, 531–555 (1927).

2. P. McMullen. “The numbers of faces of simplicial polytopes,” Israel J. Math.,9, 559–570 (1971).

3. V.N. Shevchenko. “Triangulations of convex polyhedra and their Booleanfunctions,” in: Mathematical Problems of Cybernetics, vol. 16, Fizmatlit,Moscow, 2007, pp. 43–56.

4. G. Ziegler. Lectures on polytopes, Springer, Berlin (1995).

153

Regularization and Correction Methods for ImproperConvex Programming Problems

Vladimir Skarin

Institute of mathematics and mechanics UB RAS, Ekaterinburg, Russiaskavd@imm.uran.ru

Consider the convex program (CP)

minf0(x) : x ∈ X, (P0)

where X = x : f(x) ≤ 0, f(x) = [f1(x), . . . , fm(x)] and fi(x) are convexfunctions defined on Rn, i = 0, 1, . . . ,m. Problem (P0) is called improper [1] ifthe following relations do not hold:

−∞ < f∗ = F ∗ < +∞,

where f∗ is the optimal value of problem (P0), F∗ = supλ≥0 infx L(x, λ), L(x, λ) =

f0(x) + (λ, f(x)), λ ∈ Rm+ .Depending on the sets X and Λ = λ ∈ Rm+ : infx L(x, λ) > −∞ being

empty or not there are distinquished three kinds of improper CP problems:(1) X = ∅, Λ 6= ∅; (2) X 6= ∅, Λ = ∅; (3) X = ∅, Λ = ∅. Problem of the 1st kindis encountered most often [1,2].

If the set X of problem (P0) is empty then we consider the following ap-proximation for (P0):

minf0(x) : x ∈ Xξ, (Po)ξ

where ξ = argmin‖ξ‖ : Xξ 6= ∅, Xξ = x : f(x) ≤ ξ, ξ ∈ Rm+ . In paper[2], author associates (P0) with the problem (Pσ) of finding saddle points ofthe regularized Lagrange function Lσ(x, λ) = L(x, λ) + α‖x‖2 − β‖λ‖2, whereσ = [α, β] > 0, ‖ · ‖ is the Euclidean norm. By means of this tools for theimproper problem of the 1st kind the estimates of the connection between theproblem (Pσ) and the problem (P0)ξ were established.

It is easy to see that such approach to correction of improper problemsis close to well-known regularization methods for ill-posed optimization prob-lems [3] (Tikhonov regularization, the method of quasi-solutions, the residualmethod). In the Tikhonov regularization method the following similar problemis considered along with (P0):

minf0(x) + γ‖x‖2 : x ∈ X, (P1)

where γ > 0;

154

in the method of quasi-solutions:

minf0(x) : x ∈ X ∩ Sr, (P2)

where Sr = x : ‖x‖2 ≤ r, r > 0;and in the residual method:

min‖x‖2 : x ∈ X ∩Mδ, (P3)

where Mδ = x : f0(x) ≤ δ, δ ∈ R1.The problems (P1) − (P3) are stable with respect to perturbations of the

initial data and well approximate solutions of (P0) if the latter is solvable. IfX = ∅, i.e. (P0) is improper CP problems of the 1st or the 3rd kinds then theapproach to optimal correction of improper problems can be formulated as oneof the sequences:

(P0) → (Pi) → (Pi)ξ → (Pσ),

where (Pi)ξ is constructed by analogy with (P0)ξ, i = 1, 2, 3.It is significant that the problem (P1) − (P3) can be improper ones of the

1st kind independent on an improperity class of the problem (P0). In thereport convergence conditions are formulated for the corresponding methodsand convergence estimates are given.

The author was supported by the Russian Foundation for Basic Research(projects no. 13-01-00210 and no. 13-07-00181).

References

1. I.I. Eremin, Vl.D. Mazurov, N.N. Astaf ’ev. Improper problems of linear andconvex programming, Nauka, Moscow (1983). (in Russian)

2. V.D. Skarin. “Regularized Lagrange function and correction methods for im-proper convex programming problems,” Proc. of the Steklov Institute ofMathematics, Suppl.1, 2002, S116–S144.

3. F.P.Vasil’ev. Methods of solving extremal problems, Nauka, Moscow (1981).(in Russian)

155

Optimization of the Psychodiagnostic DataProcessing Using Neural Networks

Elena Slavutskaya1, Leonid Slavutskii2

1 Chuvash State Pedagogical University, Cheboksary, Russia;slavutskayaev@gmail.com

2 Chuvash State University, Cheboksary, Russia; las co@mail.ru

The research is carried out at the intersection of computer science and psy-chology. To analyze the psychodiagnostic data the artificial neural networks(ANN) are used. The formation of the relationships between the verbal andnon-verbal intelligence of preteen children essentially depends on the emotional,volitional and communicative traits. The research is based on Cattelian Person-ality Theory. The non-verbal intelligence IQ was measured using R. B. Cattellsintelligence test and personal characteristics were tested using the R. B. Cattells12-factor questionnaire. Using ANN the computational models for dependencesof non-verbal and verbal intelligence IQ(B) are obtained. The curves at thefigure are: 1 - for children with average personal traits, 2-4 examples of IQ(B)for individual children.

From a mathematical point of view, the training of ANN is the multiparam-eter problem of nonlinear optimization. Therefore, the computational modelIQ(B) obtained for each respondent, depends on statistical sampling. The be-havior of such relationships is of great volatility, making it difficult psychologicalinterpretation. On the other hand, a statistically significant correlation betweenIQ and B is absent (see figure), because this connections are highly nonlinear.

It was found that the range L for earch curve (see figure) can be consid-ered as the quantitative psychological indicator, which is associated with certainpersonality traits much more than IQ. The correlation coefficients of the per-sonality traits with IQ and L are shown on the figure.

The neural network analysis can reveal new patterns and evaluate the sig-nificant nonlinear relationships between psychological characteristics. Such de-pendences are difficult to analyze by traditional statistical methods. The mainconclusion of the study is that for an adequate interpretation of the psychodi-agnostic results the complex using of the traditional statistical methods andartificial neural networks is optimal.

156

Fig. 1

A - gregariousness - isolation; B - abstract-concrete thinking; C - emo-tional stability - instability; D excitement - balance; E-independence - obe-dience; F carefree- concern; G - high-low discipline; H - boldness-shyness;I - softness-hardness; O - anxiety-calmness; Q3 - high-low self-control; Q4 -tension-relaxation. Note: bold correlation coefficients are of the significancelevel P < 0.025.

References

1. R.B. Cattell. Advanced in Cattelian Personality Theory. Handbook of Per-sonality. Theory and Research. The Guilford Press, N.Y. (1990).

2. E.V. Slavutskaya, L.A. Slavutskiy. “Using artificial neu-ral networks for analysis of gender differences in youngerteenagers,” Psikhologicheskie Issledovaniya, No.5 (23), (2012).http://psystudy.ru/index.php/eng/2012v5n23e/696-slavutskaya23e.html

157

Iterative procedures for solving minimax problems

Vladimir Srochko1

1 Irkutsk State University, Irkutsk, Russia; srochko@math.isu.ru

For qualitative analysis (optimality conditions), minimax problems havebeen studied well enough as separate problems and through reduction to thestandard problems with restrictions as well. In this case the corresponding opti-mality conditions are represented also in minimax form (point-wise and integralminimax principles) which makes difficult to construct effective procedures ofsuccessive improvement. Nevertheless, a few numerical methods for solving min-imax control problems in frames of necessary optimality conditions have beendeveloped, for example, in [1, 2].

In this paper two types of minimax problems – simple problem for a systemof integral functionals dependent on control only and Chebychev type problemfor standard functionals, connected with nonlinear dynamic system, are studiedin view of iterative improvement in the class of needle-shaped variations.

For solving the problem of an admissible control improvement we use a con-structive procedure of needle-shaped variation in form of a generalized convexcombination of a pair of controls. For variation parameters search we constructspecific systems of integral equations. In addition the necessary optimality con-ditions have a non-traditional form and connected with the inconsistency of theintroduced relations. For the proposed formalization it is natural to conductthe numerical solving of the auxiliary problems by the least squares method.As a result, an improvement iteration consists of solving a family of specificquadratic problems for a minimum of the Euclidean norm of the terminal statein a simple system of phase equations, where the right-hand side depends onlyon control and time.

References

1. V.F. Demyanov and others. Nonsmooth Problems of Optimization and Con-trol, Leningrad State University, Leningrad (1982).

2. V.A. Srochko. Iterative Methods for Solving Optimal Control Problems, Fiz-matlit, Moscow (2000).

158

BNBTest@HOME - A distributed implementation ofbranch and bound method

Bo Tian1, Mikhail Posypkin2

1 Lomonosov Moscow State University, Moscow, Russia;yesyestian@gmail.com

2 Lomonosov Moscow State University, Moscow, Russia; posypkin@isa.ru

Many problems from the areas of operations research and artificial intel-ligence can be defined as combinatorial optimization problems. Branch andBound (BnB) method is an universal and well known algorithmic technique forsolving problems of that type. This algorithm allows to reduce considerably thecomputation time required to explore the entire solution space associated withthe problem being solved. Often the running time remains unaffordably largeand using parallel or distributed processing is one of the major and popularways to decrease it.

The parallelization of BnB algorithms for all kinds of parallel machines wasstudied by several authors. The implementation of BnB on a shared memorysystem is relatively simple because each processor can access the global addressspace directly. The simulation of global memory on a distributed memory sys-tem for solving BnB problems was described in [1]. But these techniques areonly useful in special cases or for small numbers of processors and leads in gen-eral to communication bottlenecks. Many approaches have been proposed forimplementing BnB algorithms on tightly coupled distributed memory systems,see for example [2]. However large scale volunteer desktop grid systems was notyet considered as a target platform for branch-and-bound methods. And that’sprecisely an issue addressed in our system.

Era of volunteer computing was opened by SETI@HOME project[3]. Thisproject has proven to be one of the biggest successes of distributed computingduring the last years. With a quite simple approach SETI manages to processlarge volumes of data using a vast amount of distributed computer power. Toextend the generic usage of this kind of distributed computing tools, BOINC[4]is being developed. In this paper we propose BNBTest@HOME, an experimen-tal BOINC version to solve large-scale knapsack problem by using distributedbranch and bound method.

The BNBTest@HOME simplify large calculations by breaking them downinto many smaller ones processed by client PCs, the results of which on themaster’s side can then be recombined to generate the solution. In BNBTest theload balance is the most important issue.

159

We use breadth-first BnB in the master side, who always branches a sub-problem with minimal tier number. This technique produces as much tasksas required to be assigned to clients and generated subproblems are more uni-formly distributed that for other search strategies. Master server has to payfor this with large storage requirements. On the client side, we adopt depth-first branching strategy. Depth-first strategy uses one-sided economic branchingthat significantly reduces memory usage. Such architecture enables an efficientutilization of resources in volunteers’ computing.

This work discusses the design and deployment of distributed BnB methodbased on BOINC. How to segment task, how to promote loading balance and soon. We also evaluate and demonstrate the efficiency of the proposed approachon large-scale knapsack instances[5]. In the future we plan to try differentstrategies to compose work-units and study its impact on the efficiency of thealgorithm.

This work was supported by the China Scholarship Council (CSC).

References

1. R. Lling, B. Monien “Load Balancing for Distributed Branch&Bound Algo-rithms.”Conference Parallel Processing Symposium, Beverly Hills, 1992.

2. Yu. G. Evtushenko, V. U. Malkova, A. A. Stanevichyus “Parallel globaloptimization of functions of several variables.” CMMP, Vol. 49, No. 2,pp. 246-260.2009.

3. David P. Anderson, Jeff Cobb, Eric Korpela “SETI@home: An Experimentin Public-Resource Computing.” Communications of the ACM, Vol. 45, No.11, pp. 56-61, November 2002.

4. D. P. Anderson “Boinc: A system for public-resource computing andstorage.” Fifth IEEE/ACM International Workshop on Grid Computing,pages. 365-372, Nov. 8 2004.

5. David Pisinger “Where are the hard knapsack problems?” Computers &Operations Research, Vol. 32, No. 9, pp. 2271-2284, 2005.

160

Optimization strategies for radiation safety at lowdoses of irradiation including the cost of diagnosis of

oncologic diseases

Nikolai Tikhomirov, Tatiana Tikhomirova, Damir Ilyasov

Plekhanov Russian University of Economics, Moscow, Russia;t−tikhomirova@mail.ru

Recently, in the scientific literature, considerable attention is being paid toissues of validity of the estimates of radiation risks and risk reducing strategiesat low radiation dose (up to 100 mSv/year). This is due to the fact that overes-timation of levels of these risks because of incorrect estimation techniques leadsto increase and inefficient expenditure of funds on protection from radiation.

The radiation risk is defined in the special literature as the probability ofcancer appearance due to irradiation during subsequent years of life before theage of 70 [1]. Herewith, in areas with low doses of radiation in the absence ofreliable statistics, it is proposed to estimate his levels by linear extrapolation ofavailable data on the stochastic effects of radiation exposure to population byaverage doses (100-1000 mSv/year), that allows us to characterize these figuresas virtual. Estimates of radiation risk obtained by the latest modifications ofthe ICRP techniques for population at doses of 20 and 50 mSv/year are 10−3

and 2, 5 · 10−3 respectively.Note that the content of radiation risk is significantly different from the

concept of ”risk of vital activity ”, which usually refers to the probability of thedeath of an individual for a specific reason during the year. Levels of such risks(traffic accidents, poisoning, murder, death by disease, etc.) vary from 5 · 10−6

to 10−2 for the population in developed countries, and from 10−5 to 10−2 forprofessionals engaged in different branches of production. These values are quitestable over time and they were accepted by the society as a reference for ensuringthe safety of life in its various spheres. In this regard, it seems appropriate tolead the radiation risks to the risks of vital activity, taking into account thatthe period between the time of radiation exposure and manifestation of cancerare quite significant (from 5 to 35 years), that there is a likelihood of recoveryof a sick and the value of years of future life decreases with time.

In this context, diagnosis of cancer and subsequent medical treatment shouldbe considered as one of the most important measures in strategies to reduceradiation risks.Taking into account the cost of these activities, the objectivefunction of management of radiation safety is transformed to the form:

CR[R(Z,E0)] + CR[R(Z1, E0)] + Z1 + Z(E1 −E0) → min (1)

161

where Z1 - the costs of diagnosis and medical treatment of cancer irradiatedcontingent; Z(E1−E0) - cost of intervention reduced the dose from the level E1

to E0; CR[R(Z,E0)] - losses to society from territory pollution of the residualdose E0; CR[R(Z1, E0)] - the cost equivalent of the loss of population healthdefined with considering the costs of diagnosis, medical treatment and the levelof residual risk R(E0).

Considering inverse nature of relationships between taken into account lossesand the costs of risk reduction expressed in general form the type of depen-dencies such as CR ∼ ϕ(1/Z), the objective function (1) takes the form of aparabola, and the optimal strategy can be found as a solution to the trivial taskof searching for minimum of parameters of cost considering restrictions on theirsize and level of risk to the public.

Options for solving the optimization of strategies of radiation safety policies,obtained with considering above mentioned estimates of risks and of the criterion(1), are characterized by a significant shift allowable radiation dose E in the areaof doses 20-70 mSv/year. This is due to the reduction of 3-5 times proposedestimates of radiation risks to the public (and associated with them losses)compared to their ”virtual” values obtained by standard methods, as well asdecreasing the costs of providing security, which is a consequence of the transfermost ”expensive” measures of relocation, to the areas with higher dose levels.

These results confirm the validity of the recent recommendations of theICRP on the advisability of adoption of measures the interventions in incidentsof radiation leakage at doses greater than 20 mSv/year.

The work is supported by the grant RFBR project 13-06-0045

References

1. The 2007 Recommendation of the International Commission on RadiologicalProtection. Publication 103.// Annals of the ICRP, v. 37, 2-4. Elsevier,2007.

162

P -regularity theory and singular calculus of variationsproblems.

Alexey A. Tretyakov1

1 Dorodnicyn Computer Center, Russian Academy of Sciences, Vavilova 40,Moscow 119991, Russia; University of Siedlce, 3 Maja 54, 08-110 Siedlce,

Poland; System Research Institute, Polish Academy of Sciences, Newelska 6,01-447 Warsaw,Poland; tret@uph.edu.pl

Applications of the p-regularity theory, successfully developing for the lastyears, to calculus of variations problems are represented. The main idea ofthis theory gives a detailed description of the structure of the zero set of de-generate nonlinear mappings. Among the applications, the construction of p-factor-operator is used to construct numerical methods for solving degenerateoptimization problems and p-order necessary and sufficient conditions are for-mulated to solve constrained degenerate optimization problems, in which theLagrange multiplier associated with the objective function might be equal tozero.

Here the p-factor-approach is applied to singular calculus of variations prob-lems when constraints are nonregular at the solution, e.i. its derivatives are notsurjective. For p-regular calculus of variations problems we present necessaryoptimality conditions for optimality in singular case and illustrate our resultsby classical examples of Lagrange problems in calculus of variations.

The authors were supported by the Russian Foundation for Basic Research(project no. 11-01-00786-a by the Leading Research Schools Grant No NSH-5264.2012.1 and by the Russian Academy of Sciences Presidium Program P-18).

163

Three-Dimensional Extended Vehicle RoutingProblem

Aida Valeeva1, Yulia Goncharova,2, Ivan Koshcheev3

1 USATU, Ufa, Russia; aida val2004@mail.ru2 USATU, Ufa, Russia; yuliagonch@mail.ru

3 USATU, Ufa, Russia; ivan.koscheev.ufa@gmail.com

In this paper the problem of homogeneous product to the different customersis considered. The company, producing reagent, has the customers in differentregions of Russia, that deliver their production in required quantity by thecar heterogeneous vehicles, rented by the company. There are containers withreagent for the several customers in each vehicle. Containers with reagent arepacked on the pallets before transportation and then are located in the vehicleswith a loader. The company has a warehouse (a depot) or several warehousesfor keeping of raw products and of production produced by the company, everyroute starts and ends at the depot. The company can set a delivery period inseveral days, during which the containers with reagent must be delivered to thecustomers. It is allowed for stop in some time intervals during the containersof reagent delivery. The company is interested in increasing of benefit andin minimization of the transport costs. Three-Dimensional Extended VehicleRouting Problem (3L-EVRP), considered in this paper, belongs to the class ofNP-hard vehicle routing problems.The 3L-EVRP solving consists in solving of the following subproblems:

1. Demand of reagent forecasting.2. The delivery of the containers with reagent to the different customers.

2.1. Creation of the rational routes for the delivery of the containers withreagent by the car vehicles to the different customers subject to the followingconditions: 2.1.1. There are one or several depots for the vehicles. 2.1.2. Everyroute starts and ends at the depot. 2.1.3. A fleet of vehicles is heterogeneous.2.1.4. The time windows are considered. The service of each customer muststart within the associated time window. Moreover, in case of early arrivalat the location of customer, the vehicle generally is allowed to wait until theservice may start. 2.1.5. Delivery period of days is considered, where days isthe number of days, during which delivery of the containers with reagent mustbe carried out. 2.1.6. The period of vehicle return to the depot is determined.2.1.7. The split delivery is considered: each customer can be visited more thanone vehicle. At the same time the demand of each customer may be greaterthan the vehicle capacity and the customer may be included in several routes.2.1.8. The mass of containers with reagent, loading in vehicle, doesnt exceed

164

the capacity of the vehicle. 2.1.9. The penalty of the route, which doesntcorrespond to the rational packing of containers with reagent in vehicles duringthe delivery of the containers with reagent to the different customers on thisroute. 2.1.10. The demand of customers in reagent must be satisfied.

2.2. The selection of the containers types for the transportation in it ofreagent in vehicles (cuboids and cylinders are considered).

2.3. Packing of the containers with reagent in vehicles with the packingplan subject to the following conditions: 2.3.1. The containers with reagent arepacked on the pallets before transportation. 2.3.2. The mass of containers withreagent and pallet, loading in vehicle, doesnt exceed the capacity of the vehicle.

Mathematical model for the 3L-EVRP, prototype of the logistic transportsystem, optimization kernel are designed. For the optimization kernel the pop-ulation based ant colony optimization algorithm for solving the problem of cre-ation of the rational routes for the delivery of the containers with reagent bythe car vehicles to the different customers subject to the mentioned above con-straints and evolutionary algorithm (1+1)-EA3D for solving the packing prob-lem of the containers with reagent in the vehicles with packing plan subjectto the technological constraints are developed. The computational experimentsare resulted.

The authors were supported by the Russian Foundation for Basic Research(project no. 13-07-00579).

References

1. Archetti C., Hertz A., Speranza M.G. ”A tabu search algorithm for thesplit delivery vehicle routing problem”, Transportation Science, V. 40, P.64-73(2006)

2. Yusupova N.I., Valeeva A.F., Rassadnikova E.Yu., Koshcheev I.S., LatyipovI.M. ”Multi-criteria problem of goods delivery to the different customers”,Logistics and supply chain management, V. 5, N. 56, P. 60-82(2011)

165

On Optimal Design for Capacity and ElectricityMarket

Alexander Vasin, Polina Kartunova

Lomonosov Moscow State University, Moscow, Russia; foravas@yandex.ru

We provide an algorithm that determines the optimal capacity structurewith account of fixed and variable costs, when electricity is produced by stan-dard small generators with the same capacity and different costs. Selection ofthe optimal structure also determine the auction design that realizes. The de-mand within one annual period is price inelastic and characterized by the maxi-mum value M and the duration of load curveM(τ ). The inverse function τ (M)determines the share of the time within the period when the required capacityexceeds M . The problem is to find capacity volumes vi, i ∈ I that meet the de-mand at minimum cost. Let A be the set of units under selection, each a ∈ A ischaracterized by fixed and variable costs cfa and cva. The problem is to choose Munits ( ~a1, . . . , ~aM) that cover the demand, with minimal total costs. For a given

collection (a1, . . . , aM ) the costs are C(a1, . . . , aM ∼=∑Mi=1(c

fal+τ (l)cVal) ). An

algorithm for the optimal choice is as follows. At the first stage, find a unit thatminimizes the cost (cfa + cva) (since τ (1) = 1). Consider stage i when unitsa1, . . . , ¯ai−1 already have been determined. Find argmin

a∈A(cfa + τ (i)cva) (∗). If

there is solution a 6∈ a1, . . . , ¯ai−1 = Ai−1 then let ai = a. Otherwise, considerDLC min(M(τ ), i) obtained from the original DLC M(τ ) by limiting the max-imal demand with i units. For every unit a that is optimal in the set A/Ai−1

in the sense problem (∗) with some j ≤ i substituted for i, compute the min-imum cost necessary to cover the load min(M(τ ), i) with the set Ai−1

⋃ai :

order units in this set according to cva (from lowest to highest) and find∑

forreordered collection (a11, . . . , a1i). Denote this value by Ci(Ai−1|a). Finally,choose ai = argminCi(Ai−1|a), set Ai = Ai−1

⋃ai and pass to stage i+ 1.

Theorem 1. The set AM obtained according to the algorithm describedabove and arranged in order of increasing cva is a solution of the problem(a∗1, . . . , aM∗) = arg min

(a1,...,aM )∈AC(a1, . . . , aM ). Now consider the auction de-

sign that provides selection of the optimal capacity structure AM . Each pro-duced a can propose one unit cfa , c

va. He submits an offer (pfa, p

va). An auctioneer

processes the offers according to the above given algorithm and selects M units(a1, . . . , aM ). Each selected producer al bears the costs cfal + cval · τ (l) and ispaid pfal + pval · τ (l) + max(pfas + pvas · τ (s)− pfal + pval · τ (l)) This rule providesselection of the optimal capacity structured under the general condition.

166

Algorithms for maxmin location problem on a plane

Gennady Zabudsky1, Anton Koval2

1 Omsk Branch of Sobolev Institute of Mathematics SB RAS, FinancialUniversity, Omsk, Russia; zabudsky@ofim.oscsbras.ru

2 Omsk Branch of Sobolev Institute of Mathematics SB RAS, Omsk, Russia;akoval@thumbtack.net

Maxmin criteria is often used in location problems of undesirable facilities.Facility is called undesirable if it may pose a danger to an individual livingnearby. Examples of undesirable facilities are nuclear power plants, chemicalplants and garbage dumps. Such facilities should be placed as far as possiblefrom residential areas.

Let S is rectalinear region on the plane; V = vi|i ∈ N is a set of points(called demand points) in S; O = oj |j ∈ P is a set of undesirable facilities; Rijis the minimum distance between vi and oj ;M is the minimum distance betweenfacilities; wi > 0 is a weight of vi; ρ is the Chebyshev metric. The objectiveis to maximize the minimum weighted distance between demand points andfacilities.

The maxmin model takes the following form:

L→ max, (1)

ρ(oj , vi) ≥ max(Rij,L

wi), i ∈ N, j ∈ P, (2)

ρ(oi, oj) ≥M, i, j ∈ P, (3)

oj ∈ S, j ∈ P. (4)

The algorithm of approximate solutions to a given accuracy for problem (1)– (4) is proposed. The experiment with the algorithm and IBM ILOG CPLEXusing integer programming model is implemented.

The combinatorial algorithm of finding an exact solution for special cases ofthe problem is developed.

The authors were supported by the Russian Foundation for Basic Research(project no. 13-01-00862).

References

1. M.J. Katz, K. Kedem, M. Segal “Improved algorithms for placing undesirablefacilities,” Computers & Operations Research, No. 29, 1859–1872 (2002).

167

The development of specialized computer knowledgecontrol system using discrete optimization models

L.A. Zaozerskaya1, V.A. Plankova1

1 Omsk Branch of Sobolev Institute of Mathematics of the Siberian Branch ofthe RAS, Omsk, Russia;

zaozer@ofim.oscsbras.ru, plankova@ofim.oscsbras.ru

The paper deals with aspects of creation of specialized computer testingsystems. System EMM test is considered as an example. It was designed bythe authors for knowledge control to the course ”Mathematical methods inEconomics”.

One of the important problems at development of such systems is the deter-mination of the test content. To solve it, the method based on the formalizationof the concept of the test optimal structure has been proposed. In case of lim-ited time such test provides a fairly objective estimation of student’s knowledge.Models of integer linear programming to determine the test optimal structurewere proposed [1]. System EMM test was designed on the basis of one of thesemodels. The process of test formation is automated. It is based on algorithmsdeveloped by the authors to generate variants of test tasks in the on-line mode.

In case of large volume training course a single structure can don’t ensurethe sufficient completeness of knowledge control of all it’s elements. Thereforeto estimate mastering of the discipline by a group of students the using of a setof test structures is needed. We proposed mathematical models to determine ofan optimal set of test structures.

The structure and the diagram of functioning of the computer systemEMM test will be considered. Results from use of the system in training ofstudents with economics specialties will be demonstrated. Areas of furtherdevelopment of the system as well as prospects of such approach in creatingspecialized computer testing systems for other disciplines will be discussed.

References

1. L.A. Zaozerskaya, V.A. Plankova. “Application of Discrete OptimizationModels in Development of an Automatic Knowledge Control System”,Vestnik NGU, seriya “Informatcionnye tehnologii”, 6, No. 1, 47–52 (2008).

168

Determining parameters in the model of watertransfer in soil

E.S. Zasukhina, V.V. Dikusar

Computing Center RAS, Moscow, Russia; zasuhina@ccas.ru

A process of vertical water transfer in soil can be described by one-dimensionalnonlinear parabolic equation. Consider following initial-boundary value prob-lem:

∂θ

∂t=

∂

∂z

(D(θ)

∂θ

∂z

)− ∂K

∂z, (z, t) ∈ Q,

θ(z, 0) = ϕ(z), z ∈ (0,L),θ(L, t) = ψ(t), t ∈ (0, T ),∂θ

∂t

∣∣∣∣z=0

=

(D(θ)

∂θ

∂z−K

)∣∣∣∣z=0

+R(t)− E(t), t ∈ (0, T ),

θmin ≤ θ(0, t) ≤ θmax, t ∈ (0, T ).

(1)

Here, z is the coordinate; t is time; θ(z, t) is humidity at point (z, t); Q =(0, L)× (0, T ); ϕ(z) and ψ(t) are given functions; D(θ) is the coefficient of diffu-sion; K(θ) is the hydraulic conductivity; R(t) is precipitation; E(t) is evapora-tion; θmin and θmax are minimal and maximal values of humidity respectively.According to [1] D(θ) and K(θ) are given in the form:

K(θ) = K0S0.5[1− (1− S1/m)m]2,

D(θ) = K01−m

αm(θmax − θmin)S0.5−1/m ×

×[(1− S1/m)−m + (1− S1/m)m − 2],

where S =θ − θmin

θmax − θmin, K0, α are given coefficients, and θmin, θmax, m are

some unknown parameters.These parameters can be determined by solving some optimal control prob-

lem. Suppose that a function θ(z, t) is defined in Q0 ⊂ Q and the set Q0 consistsof finite number of elements q, q = (z, t). This function can be interpreted asthe experimental data. Consider following optimal control problem: determineuopt = (θoptmin, θ

optmax,m

opt) and corresponding optimal solution θopt(z, t) such

that functional J(u) = 12

∑q∈Q0

[θ(q)− θ(q)]2 is minimized.

169

The problem (1) is discretized by following finite-difference scheme:

θn+1i − θni

τ=

1

h

(Dni+1/2

θn+1i+1 − θn+1

i

h−Kn

i+1/2−

− Dni−1/2

θn+1i − θn+1

i−1

h+Kn

i−1/2

),

1 < i < I, 0 ≤ n < N,

where θni ,Dni+1/2,K

ni−1/2 are values of the functions θ(z, t),D(θ(z, t)),K(θ(z, t))

at the points (ih, nτ ), ((i+1/2)h, nτ ), ((i− 1/2)h, nτ ) respectively. On the leftboundary

θn+10 − θn0

τ=

2

h

(Dn

1/2

θn+11 − θn+1

0

h−Kn

1/2 +Rn+1 − En+1

),

0 ≤ n < N.

Here Rn+1, En+1 are values of R(t) and E(t) at t = (n+ 1)τ .Suppose that Q0 = (ih, nτ ), 1 < i < I, 0 ≤ n < N.Then the discrete optimization problem is solved by gradient method basing

on fast differentiation formulas [2].The problem was solved with concrete values of input parameters. The nu-

merical experiments lead to following conclusion: two parameters θmin, θmaxare determined with high accuracy under the condition that the third parameterm is known. As to the case when all three parameters are to be determinedoptimization process was difficult because of the structure of the objective func-tion. In this case the solution was determined by gully method but with lesseraccuracy.

The author was supported by the Russian Foundation for Basic Research(project no. 11-01-00786-a, project no. 11-07-00201-a, project no. SS-5264.2012.1).

References

1. M.Th. Van Genuchten. “A closed form equation for predicting the hydraulicconductivity of unsaturated soils,” Soil. Sci. Soc. Am. J., 44, 892–898(1980).

2. Yu.G. Evtushenko. “Computation of exact gradients in distributed dynamicsystems,” Optimization methods and software, 9, 45–75 (1998).

170

Two-stage affine scaling methods for linearsemidefinite programming problem

Vitaly Zhadan1

1 Dorodnicyn Computing Centre of RAS, Moscow, Russia; zhadan@ccas.ru

We consider the linear semidefinite programming problem in standard form

min C •X, Ai •X = bi, 1 ≤ i ≤ m, X 0, (1)

where C,X and Ai, 1 ≤ i ≤ m, are symmetric matrices of order n, the inequalityX 0 indicates that X must be a semidefinite matrix. The operator • denotesthe standard inner product between two matrices.

For solving (1) and the corresponding dual problem the primal and dualaffine-scaling methods have been proposed [1,2]. These methods under ordinaryassumptions concerning nondegeneracy and strict complementarity converge lo-cally to solutions of these problems. But step sizes applying in these methodswere taken constant and sufficiently small.

In this paper we propose to use the steepest descent approach for choosingthe step sizes, especially on the first stage of calculations, when current pointsare far from solutions. Since in this case current points may belong to boundariesof the feasible sets, the special modifications of right hand sides in iterativeprocesses are considered. The relation between these variants of affine-scalingmethods and simplex-type methods is also discussed

The author was supported by the Programs of Fundamental Research ofRussian Academy of Sciences P-15, by the Russian Foundation for Basic Re-search (project no.11-01-00786), and by the Leading Scientific Schools Grantno.5264.2012.1

References

1. M.S. Babynin, V.G. Zhadan. “A Primal Interior Point Methods for the Lin-ear Semidefinite Programming Problem,” Computational Mathematics andMathematical Physics, 48, No. 10, 1746–1767 (2008).

2. V.G. Zhadan, A.A. Orlov. “Dual Interior Point Methods for Linear Semidef-inite Programming Problems,” Computational Mathematics and Mathemat-ical Physics, 51, No. 12, 2031–2051 (2011).

171

On the New Statement of the Optimal ControlProblem of the Metal Crystallization Process

Vladimir Zubov1, Alla Albu2

1 Dorodnicyn Computing Centre of RAS, Moscow, Russia; zubov@ccas.ru2 Dorodnicyn Computing Centre of RAS, Moscow, Russia; alla.albu@mail.ru

The optimal control of metal solidification in the mold for an improved setupis studied. The mold and the liquid metal in it are cooled in a special setupconsisting of a melting furnace and a cooler. The problem of optimal controlconsists in choosing a regime of metal cooling in which the solidification fronthas a preset shape. The speed of the mold in the furnace is used as the controlfunction.

This problem has previously been solved in the case of a simple model of themelting furnace consisting of two vertical walls. The numerical solution of theproblem of optimal control was obtained for a wide range of input parameters(temperature of the furnace, the maximum allowed depth of the immersion ofthe object into the cooler, etc.). These parameters have a significant impacton the evolution of the crystallization process. However, for the achievementof a significant improvement in the quality of the obtained metal other setupparameters are also significant.

In the new statement of the optimal control problem a more advanced modelof the furnace and its impact on the metal is used. The side vertical walls arejoined now on the top with a horizontal wall (the roof), which is heated up toa certain temperature. Two side walls of the mold, which are perpendicular tothe heated walls of the furnace, are heat insulated now. For the new model hasbeen performed a series of calculations of the direct problem. The results ofthe calculations have shown that even during the ”worst” scenario there is noformation of ”bubbles” with liquid metal inside the object (unlike when con-sidering the previous model). The optimal control problem in a new statementis solved for an object of the simplest form - the parallelepiped. Studies haveshown that in this case the interface phase rectified considerably more thanwhen the previous simple model is considered.

This work was supported by the RFBR (project No. 12-01-00572-), bythe Program for Fundamental Research of Presidium of RAS P18, and by theProgram ”Leading Scientific Schools” (NSh-5264.2012.1).

172

Efficiency of parallel computing in extragradientmethods

Anna Zykina, Dmitry Zaporozhets, Dmitry Kujanov, VladimirZykin

Omsk state technical university, Omsk, Russia; avzykina@mail.ru

It’s investigated the efficiency of parallel computing in one-step and two-stepextragradient methods. Estimates of the minimum dimension of the problemand the optimal number of processors to achieve maximum acceleration. Thepresented theoretical estimates of the convergence acceleration are confirmedby numerical experiments.

For solutions variation inequalities 〈H(z∗), z − z∗〉 ≥ 0, ∀z ∈ Ω, considerone-step extragradient method [1, 2]

xk = PΩ(xk − αkH(xk)), xk+1 = PΩ(xk − αkH(xk)),

and two-step extragradient method [3, 4]

xk = PΩ(xk − αkH(xk)), xk = PΩ(xk − αkH(xk)),

xk+1 = PΩ(xk − αkH(xk)).

Here H(z) : Rn → Rn,Ω - convex closed set of Rn, PΩ – operation of projectionon the set Ω, αk - size of the step methods, xk, xk – auxiliary point of method.

We analyze the acceleration of the test algorithms for solving systems of lin-ear algebraic equations, linear complementarity problems, nonlinear variationalinequalities on test examples from the I.V. Konnov’s monograph [5].

For the one-step and two-step extragradient method for solving systems oflinear algebraic equations on p processors acceleration obtained from paralleliza-tion is given by

S1(n, p) =p(4n2 + 6n)

4n2 + 6n+ 4000p(p − 1)

and

S2(n, p) =p(6n2 + 9n)

6n2 + 9n+ 6000p(p − 1)

respectively. Solving inequalities S1(n, p) > 1 S2(n, p) > 1 with respect n , weobtain for a given number of processors p estimate the minimum dimension ofthe problem, at which the acceleration is observed

n >

[√9 + 16000p − 3

4

].

173

Thus, when it’s parallelized algorithms for the one-step and two-step ex-tragradient method for solving systems of linear algebraic equations on a dual-processor computer, the acceleration will be observed at n > 44.

Analysis of the acceleration for solving linear complementarity problems andnonlinear variational inequalities respectively is given by

n >

[√25 + 16000p − 5

4

], n >

[2000p

7

].

To verify the theoretical results the program were written in an object ori-ented programming language C + + using OpenMP technology for the devel-opment of parallel programs using multiple processors with shared memory.Numerical experiments were performed on a dual-processor machine. For eachdimension of the problem carried out a series of experiments. Achieved practi-cal acceleration was calculated by averaging the value of the acceleration of theresulting sample.

The authors were supported by the Russian Foundation for Basic Research(project no 12-01-31360, 12-07-00326).

References

1. A.S. Antipin. “About the method for solution of convex programming usinga simmetric modification of the Lagrangian Function,” Economics and themathematical methods, V. 12, No. 6, 1164–1173 (1976).

2. G.M. Korpelevich. “The extragradient methof of finding of suddle points andof other,” Economics and the mathematical methods, V. 12, No. 4, 747–756(1976).

3. A.V. Zykina, N.V. Melenchuk. “ The two-step extragradient method for thevariational inequalities,” Izvestia VUZov. News of the High Mathematics,No. 9, 82–85 (2010).

4. D.N. Zaporozhets, A.V. Zykina, N.V. Melenchuk. “ Comparative analysis ofthe extragradient methods for solution of the variational inequalities of someproblems,” Automation and remote control, V. 73, No. 4, 626–636 (2012).

5. I.V. Konnov. “Combined relaxation methods for variational inequalities,Springer, Berlin (2001).

174

On Complexity of Buffers Allocation Problems forProduction Lines with Unreliable Machines

Alexandre Dolgui1, Anton Eremeev2, Mikhail Kovalyov3,Vyacheslav Sigaev2

1 Ecole Nationale Superieure des Mines, Saint-Etienne, France;dolgui@emse.fr

2 Omsk Branch of Sobolev Institute of Mathematics, Omsk, Russia;eremeev@ofim.oscsbras.ru

3 United Institute of Informatics Problems, Minsk, Belaruskovalyov my@newman.bas-net.by

Buffer capacity allocation problems for flow-line manufacturing systems withunreliable machines are studied. These problems arise in a wide range of man-ufacturing systems and concern determining buffer capacities with respect to agiven optimality criterion which can depend on the average production rate ofthe line, buffer cost, inventory cost, etc.

In our previous publications [1,2], two configurations of flow lines were con-sidered: the tandem lines where machines are in series and the lines with series-parallel machines. The present work deals with the computational complexityof buffer allocation problem for these two line configurations. The problem isproven to be NP-hard for tandem production lines with oracle representation ofthe revenue and cost functions, and it is proven to be NP-hard for series-parallellines with stepwise revenue functions.

The authors were supported by the Russian Foundation for Basic Research(projects no. 07-01-00410 and 12-01-00122), by Presidium RAS and SB RASprojects.

References

1. A. Dolgui, A.V. Eremeev, A.A. Kolokolov, V.S. Sigaev “A genetic algorithmfor allocation of buffer storage capacities in production line with unreliablemachines,” Journal of Mathematical Modelling and Algorithms, 2, 89–104(2002).

2. A. Dolgui, A.V. Eremeev, V.S. Sigaev “HBBA: hybrid algorithm for bufferallocation in tandem production lines,” Journal of Intelligent Manufacturing,18, No. 3, 411–420 (2007).

175

Multi-objective generalization of probability ofimprovement method

Alexey M. Nazarenko1, Alexis I. Pospelov2, Fedor V. Gubarev3

1 Datadvance llc., Moscow, Russia; alexey.nazarenko@datadvance.net2 Datadvance llc., Moscow, Russia; alexis.pospelov@datadvance.net3 Datadvance llc., Moscow, Russia; fedor.gubarev@datadvance.net

Surrogate-based optimization is a prominent approach to deal with compu-tationally intensive models. Time considerations become especially pressing ina multi-objective case. In this work we propose a new multi-objective optimiza-tion algorithm based on a generalization of probability of improvement method(developed in [1]).

Key component of suggested algorithm is a certain scalarization approach.Scalarization itself has a mixed reputation in a field. Single obvious advantage –possibility to apply a single-objective method, is usually considered outweighedby numerous problems, crucial ones being non-uniform frontier discovery andwasteful evaluations. However, there is a way around both of these limitations.

Frontier discovery can be improved by adaptive selection of scalarizationparameters. Our incrementally growing knowledge about Pareto-frontier allowsus to direct search into the most promising areas.

Surrogate models help us to tie together usually independent scalarized sub-tasks. All expensive evaluations made during solution of prior subtasks are usedto build more and more precise models thus making current subtask incremen-tally easier.

References

1. H. Kushner “A new method of locating the maximum point of an arbitrarymultipeak curve in the presence of noise,” Journal of Basic Engineering, 86,97–106, 1964.

176

Multicriteria optimization technique based on localgeometry of Pareto set

Nikita Perestoronin1, Fedor Gubarev2, Alexis Pospelov3

1 Datadvance LLC, Moscow, Russia; nikita.perestoronin@datadvance.net2 Datadvance LLC, Moscow, Russia; fedor.gubarev@datadvance.net

3 Datadvance LLC, Moscow, Russia; alexis.pospelov@datadvance.net

We present an algorithm for nonlinear multicriteria constrained optimizationwhich allows to find a single optimal solution or produce sufficiently dense setof Pareto optimal solutions covering the actual Pareto frontier. The underlyingidea of the algorithm is to maximally avoid functions evaluations away fromPareto-frontier.

For the first task (finding single solution) it uses generalizations of steepestdescent and quasi-Newton methods to the multi-objective case. For the wholePareto frontier to be discovered it executes the following operations: startingwith some optimal point, it makes step in design space along tangent to Paretoset and then push obtained (generically not optimal) point to optimality solvingthe first task. Repeating this procedure from the next optimal point it step-by-step discover the Pareto-frontier. Question of how to choose step length inorder to cover Pareto-frontier uniformly is considered. Also, second-order localapproximaltion of Pareto set is suggested to make steps more effective.

The algorithm is designed to efficiently solve engineering optimization prob-lems where main difficulty is the expensiveness of objectives and constraintsevaluations.

References

1. J.Fliege, B.F.Svaiter. ”Steepest descent methods for multicriteria optimiza-tion,” Mathematical Methods of Operations Research, 51, No. 3, 479–494(2000).

2. J.Fliege, L.M.G. Drummond, B.F.Svaiter. ”Newton’s method for multiob-jective optimization,” SIAM Journal on Optimization, 20, No. 2, 602–626(2009).

177

Author index

Abdullaev Vagif . . . . . . . . . . . . . . . . . . 11

Abgaryan K.K. . . . . . . . . . . . . . . . . . . . 13

Abramov Aleksandr . . . . . . . . . . . . . . 78

Abramov Alexander P. . . . . . . . . . . . 15

Abramov Oleg V. . . . . . . . . . . . . . . . . 16

Afanasyev Alexander . . . . . . . . . . . . . 18

Afanasyeva Lubov . . . . . . . . . . . . . . . . 98

Aida-zade Kamil . . . . . . . . . . . . . . . . . 20

Albu Alla . . . . . . . . . . . . . . . . . . . . . . . 172

Andrianov Alexander . . . . . . . . . . . . . 22

Anikin Anton . . . . . . . . . . . . . . . . . . . . 22

Anop Maxim . . . . . . . . . . . . . . . . . 24, 26

Antipin Anatoly . . . . . . . . . . . . . . . . . . 27

Ashrafova Yegana . . . . . . . . . . . . . . . . 29

Astrakov Sergey . . . . . . . . . . . . . . . . . . 31

Atamuratov A.Z. . . . . . . . . . . . . . . . . . 32

Avvakumov Sergey . . . . . . . . . . . . . . . 96

Balashov Maxim . . . . . . . . . . . . . . . . . 34

Buldaev A.S. . . . . . . . . . . . . . . . . . . . . . 36

Bushenkov Vladimir . . . . . . . . . . . . . . 38

Chukanov Sergey . . . . . . . . . . . . . . . . . 39

Denisov Dmitry . . . . . . . . . . . . . . . . . . 40

Dikusar V.V. . . . . . . . . . . . . 42, 44, 169

Dolgui Alexandre . . . . . . . . . . . . . . . 175

Dorjieva Anna . . . . . . . . . . . . . . . . 46, 72

Druzhinina Olga . . . . . . . . . . . . . . . . . . 47

Dvure henskiy Pavel . . . . . . . . . . . . . 49

Elkin Vladimir . . . . . . . . . . . . . . . . . . . 51

Eremeev Anton . . . . . . . . . . . . . . . . . 175

Eremin Ivan . . . . . . . . . . . . . . . . . . . . . . 52

Erzin Adil . . . . . . . . . . . . . . . . . . . . . . . . 54

Evdonov Georgy . . . . . . . . . . . . . . . . . 55

Evtushenko Yury . . . . . . . . . . . . . . . . . 57

Faizullin Rashid . . . . . . . . . . . . . . . . . . 58

Ferreira Margarida . . . . . . . . . . . . . . . 38

Finkelstein Evgeniya . . . . . . . . . . . . . 74

Førsund Finn . . . . . . . . . . . . . . . . . . . 102

Gaeva Zuhura . . . . . . . . . . . . . . . . . . . . 59

Galashov Alexander . . . . . . . . . . . . . . 61

Galiev Shamil . . . . . . . . . . . . . . . . . . . . 62

Gasnikov Alexander . . . . . . . . . . . . . . 64

Gimadi Edward . . . . . . . . . . . . . . . . . . 52

Golikov Alexander . . . . . . . . . . . . . . . . 66

Golshteyn Evgeny . . . . . . . . . . . . . . . . 67

Golubev Maxim . . . . . . . . . . . . . . . . . . 68

Gon harov Evgeniy . . . . . . . . . . . . . . . 70

Gon harova Yulia . . . . . . . . . . . . . . . 164

Gornov Alexander . . . . . . . . 22, 72, 74

Gorokhovik Valentin . . . . . . . . . . 75, 76

Grebenyuk Yana . . . . . . . . . . . . . . . . . 78

Gubarev Fedor V. . . . . . . . . . . 176, 177

Gusyatnikova P.P. . . . . . . . . . . . . . . . . 42

Husnullin Nail . . . . . . . . . . . . . . . . . . . 105

Ilyasov Damir . . . . . . . . . . . . . . . . . . . 161

Ja imovi Miloji a . . . . . . . . . . . . . . . 79

Ja imovi Vladimir . . . . . . . . . . . . . . . 80

Kaporin Igor . . . . . . . . . . . . . . . . . . . . . 81

Karpenko Anatoly . . . . . . . . . . . . . . . . 83

Karpy hev Alexey . . . . . . . . . . . . . . . 143

Kartunova Polina . . . . . . . . . . . . . . . 166

Katueva Yaroslava . . . . . . . . . . . . . . . . 24

Kel'manov Alexander . . 52, 61, 8589

Kha haturov Ruben.V. . . . . . . . . . . . 90

Kha hay Mi hael . . . . . . . . . . . . . 52, 92

Khamidullin Sergey . . . . . . . . . . . . . . 85

Khandeev Vladimir . . . . . . . . . . . . . . . 86

Khoroshilova Elena . . . . . . . . . . . . . . . 94

Kiselev Yuri . . . . . . . . . . . . . . . . . . . . . . 96

178

Kolokolov Alexander . . . . . . . . . . . . . 98

Konovalova Ekaterina . . . . . . . . . . . . 99

Kosh heev Ivan . . . . . . . . . . . . . . . . . 164

Koterov Vladimir N. . . . . . . . . . . . . 101

Koval Anton . . . . . . . . . . . . . . . . . . . . 167

Kovalyov Mikhail . . . . . . . . . . . . . . . . 175

Krivonozhko Vladimir . . . . . . . . . . . 102

Krivtsov Vladimir . . . . . . . . . . . . . . . 103

Kujanov Dmitry . . . . . . . . . . . . . . . . 173

Lazarev Alexander . . . . . . . . . 105, 143

Lemtyuzhnikova Daria . . . . . . . . . . 106

Lisana Maria . . . . . . . . . . . . . . . . . . . . 62

Lotov Alexander . . . . . . . . . . . . . . . . 108

Ly hev Andrey . . . . . . . . . . . . . . . . . . 102

Makarova Maria . . . . . . . . . . . . . . . . . 109

Maksimov Vya heslav . . . . . . . . . . . 111

Malkov Ustav . . . . . . . . . . . . . . . . . . . . 67

Malkova V.U. . . . . . . . . . . . . . . . . . . . 112

Masina Olga . . . . . . . . . . . . . . . . . . . . 114

Mijajlovi Nevena . . . . . . . . . . . . . . . . 79

Mikhailov I.E. . . . . . . . . . . . . . . . . . . . . 32

Mikhailova Liudmila . . . . . . . . . . . . . . 87

Milyukova Olga . . . . . . . . . . . . . . . . . . 81

Mizhidon Arsalan . . . . . . . . . . . . . . . 116

Mizhidon Klara . . . . . . . . . . . . . . . . . 116

Mol hanov Evgeny . . . . . . . . . . . . . . 118

Morozov Yury . . . . . . . . . . . . . . . . . . . 120

Murashkin Evgenii . . . . . . . . . . 26, 122

Muravey L.A. . . . . . . . . . . . . . . . . . . . . 32

Musatova Elena . . . . . . . . . . . . . . . . . 105

Nazarenko Alexey M. . . . . . . . . . . . 176

Nazarov Dmitry . . . . . . . . . . . . . . . . . 125

Nedi Angelia . . . . . . . . . . . . . . . . . . . 127

Nekrasova Olga . . . . . . . . . . . . . . . . . . 40

Nesterov Yurii . . . . . . . . . . . 49, 64, 128

Obradovi Lazar . . . . . . . . . . . . . . . . 129

Olenev Ni holas . . . . . . . . . . . . . 44, 109

Panyukov Anatoly . . . . . . . . . . . . . . . . 99

Parkhomenko Valeriy . . . . . . . . . . . . 130

Pesterev Alexander . . . . . . . . . . . . . . 132

Petrov Lev F. . . . . . . . . . . . . . . . . . . . 134

Petrova Natalia . . . . . . . . . . . . . . . . . . 47

Plankova V.A. . . . . . . . . . . . . . . . . . . 168

Plyasunov Alexander . . . . . . . . . . . . 136

Popov Leonid . . . . . . . . . . . . . . . . . . . 138

Perestoronin Nikita . . . . . . . . . . . . . 177

Pospelov Alexis . . . . . . . . . . . . 176, 177

Pospelov I.G. . . . . . . . . . . . . . . . . . . . . 140

Posypkin Mikhail . . . . . . . . 57, 83, 160

Putilina Elena . . . . . . . . . . . . . . . . . . . . 18

Pyatkin Artem . . . . . . . . . . . 52, 89, 141

Rahimov Anar . . . . . . . . . . . . . . . . . . . 20

Ribeiro Ana . . . . . . . . . . . . . . . . . . . . . . 38

Roman henko Semyon . . . . . . . . . . . . 88

Rubtsov Anton . . . . . . . . . . . . . . . . . . . 83

Sadykov Ruslan . . . . . . . . . . . . . . . . . 143

Sakharov Maxim . . . . . . . . . . . . . . . . . 83

Savelyev Vladimir . . . . . . . . . . . . . . . 145

Serovajsky Simon . . . . . . . . . . . . . . . 146

Shakenov Ilyas . . . . . . . . . . . . . . . . . . 148

Shakenov Kanat . . . . . . . . . . . . . . . . . 150

Shananin Alexander . . . . . . . . . 59, 118

Sh herbina Oleg . . . . . . . . . . . . . . . . . 106

Shev henko Valery . . . . . . . . . . . . . . 152

Sigaev Vya heslav . . . . . . . . . . . . . . . 175

Skarin Vladimir . . . . . . . . . . . . . . . . . 154

Slavutskaya Elena . . . . . . . . . . . . . . . 156

Slavutskii Leonid . . . . . . . . . . . . . . . . 156

Smirnov Georgi . . . . . . . . . . . . . . . . . . 38

Sokolov Nikolay . . . . . . . . . . . . . . . . . . 67

Soloviev Vi tor . . . . . . . . . . . . . . . . . . 103

Sro hko Vladimir . . . . . . . . . . . . . . . . 158

Sviridenko Aleksander . . . . . . . . . . . 106

Tian Bo . . . . . . . . . . . . . . . . . . . . . . . . . 159

Tikhomirov Nikolai . . . . . . . . . . . . . . 161

Tikhomirova Tatiana . . . . . . . . . . . . 161

179

Tramovi h Marina A. . . . . . . . . . . . 76

Tretyakov Alexey A. . . . . . . . . . . . . 163

Valeeva Aida . . . . . . . . . . . . . . . . . . . . 164

Vasin Alexander . . . . . . . . . . . . . . . . . 166

Wojtowi z M. . . . . . . . . . . . . . . . . . . . . 44

Zabudsky Gennady . . . . . . . . . . . . . . 167

Zaozerskaya L.A. . . . . . . . . . . . . . . . . 168

Zaporozhets Dmitry . . . . . . . . . . . . . 173

Zasukhina E.S. . . . . . . . . . . . . . . . . . . 169

Zhadan Vitaly . . . . . . . . . . . . . . . . . . 171

Zubov Vladimir . . . . . . . . . . . . . . . . . 172

Zykin Vladimir . . . . . . . . . . . . . . . . . . 173

Zykina Anna . . . . . . . . . . . . . . . . . . . . 173

180