Semi-Infinite and Robust Optimization

Elements of Semi-Infinite and Robust Optimization

Motivatio

4th International Summer SchoolAchievements and Applications of Contemporary Informatics, Mathematics and PhysicsNational University of Technology of the UkraineKiev, Ukraine, August 5-16, 2009

GerhardGerhardGerhardGerhardGerhardGerhardGerhardGerhard--------Wilhelm Weber Wilhelm Weber Wilhelm Weber Wilhelm Weber Wilhelm Weber Wilhelm Weber Wilhelm Weber Wilhelm Weber *,*, Başak AktekeBaşak AktekeBaşak AktekeBaşak AktekeBaşak AktekeBaşak AktekeBaşak AktekeBaşak Akteke--------Öztürk Öztürk Öztürk Öztürk Öztürk Öztürk Öztürk Öztürk

Institute of Applied Mathematics Institute of Applied Mathematics Programs of Financial Mathematics, Actuarial Sciences and Scientific ComputingPrograms of Financial Mathematics, Actuarial Sciences and Scientific Computing

Department of Biomedical EngineeringDepartment of Biomedical EngineeringMiddle East Technical University, Ankara, TurkeyMiddle East Technical University, Ankara, Turkey

* * * * * * * * Faculty of Economics, Management and Law, University of Siegen, GermanyFaculty of Economics, Management and Law, University of Siegen, GermanyCenter for Research on Optimization and Control, University of Aveiro, Portugal

n

GENE time 0 9.5 11.5 13.5 15.5 18.5 20.5

'YHR007C' 0.224 0.367 0.312 0.014 -0.003 -1.357 -0.811

'YAL051W' 0.002 0.634 0.31 0.441 0.458 -0.136 0.275

'YAL054C' -1.07 -0.51 -0.22 -0.012 -0.215 1.741 4.239

'YAL056W' 0.09 0.884 0.165 0.199 0.034 0.148 0.935

'PRS316' -0.046 0.635 0.194 0.291 0.271 0.488 0.533

'KAN-MX' 0.162 0.159 0.609 0.481 0.447 1.541 1.449

'E. COLI #10' -0.013 0.88 -0.009 0.144 -0.001 0.14 0.192

Networks and Optimization

'E. COLI #10' -0.013 0.88 -0.009 0.144 -0.001 0.14 0.192

'E. COLI #33' -0.405 0.853 -0.259 -0.124 -1.181 0.095 0.027

ex.: yeast data

min( ), ( ), ( )ij i im c d∗ ∗ ∗

l

( 1, . . . , )( , ) ( )n

j np m y yα∗ =≤∑

subject to

)2

1

0

l

M E C E D Eαα ακκ κ

α

∗ −∗ ∗ ∗

= ∞

+ + −∑ &

GSIP relaxation

Networks and Optimization

set of combined environmental effects combined environmental effects combined environmental effects combined environmental effects

( , ) :Y C D∗ ∗ =

1,..., 1,...,1,...,

( 0, ) ( 0, )i ii n i n

m

c d∗ ∗

= ==

× ∏ ∏l

l

1

1

1

, m in

( 1, . . . , )

( 1, . . . , )

( , ) ( )

( , ) ( )

( , ) ( )

&

i j i j ji

n

i ii

n

i ii

i i i

j n

m

p m y y

q c y y

d y y

m

α

β

ζ γ

δ

∗

=

∗

=

∗

=

=

=

≤

≤

≤

≥

∑

∑

∑

l l l l

o v e ra ll b o x c o n s t ra in ts

( ( , ))y Y C D∗ ∗∈

( 1, . . . , )i n=

Generalized Semi-Infinite Optimization

2C

I, K, L finite

Semi-Infinite Optimization

Hubertus Th. JongenSemi-Infinite Optimization,EURO XXIII 2009, July 5-9, 2009, Bonn, Germany

Semi-Infinite Optimization

Semi-Infinite Optimization

Generalized Semi-Infinite Optimization

)(τψτ

ψ

Generalized Semi-Infinite Optimization

structurally stable

global local global

)(⋅ε

nIR

asymptotic

effect

homeom.

),( τϕ ⋅

Thm. (W. 1999/2003, 2006):

⇔ξ

Generalized Semi-Infinite Optimization

constructions

Morse theory, topologymax-type, nonsmooth functions

Generalized Semi-Infinite Optimization

given

feasible set

perturbed

perturbed

given

nonsmooth GSIP

r R

B

time-minimal cooling (or heating) of

Generalized Semi-Infinite Optimization

!0 ∃>∀T

GSIP

further ex. : thermo-regulation of premature infants

control of global warming

•

•

Generalized Semi-Infinite Optimization

anticipation

maximization of time-horizon longest term description•

Hoffmann, Reinhard

Ex.: approx. of a thermo-couple characteristic

thermo-couple f (y) : spline of polynomials with deg. 3 – 13, on [a,b]

to be approx. by :

bounds on error•

Generalized Semi-Infinite Optimization

Bernhard

some interpol.

(= y)•

Ex.: approx. of a thermo-couple characteristic

thermo-couple f (y) : spline of polynomials with deg. 3 – 13, on [a,b]

to be approx. by :

bounds on error•

Generalized Semi-Infinite Optimization

some interpol.

•

Ex.: approx. of a thermo-couple characteristic

thermo-couple f (y) : spline of polynomials with deg. 3 – 13, on [a,b]

to be approx. by :

bounds on error•

Generalized Semi-Infinite Optimization

some interpol.

•time

by discretization

43421 {

Generalized Semi-Infinite Optimization

numerical methods

parametrically,by approximation

Generalized Semi-Infinite Optimization

numerical methods

by local linearization & transversal intersection

1

3

Generalized Semi-Infinite Optimization

numerical methods

reduction ansatzexchange methodsemismooth Newton’s method

O. Stein, G. Still W.A. Tezel

Robust Optimization Laurent El Ghaoui

.

Robust Optimizationand Applications,IMA Tutorial, March 11, 2003

LP as a conic problem

Robust Optimization

.

.

.

CQP

.

.

Robust Optimization

.

.

semidefinite programming (SDP)

.

.

Robust Optimization

CQP .

.

.

dual of conic program

.

Robust Optimization

,

.

.

robust conic programming

Robust Optimization

.

.

.

.

robust conic programming

Robust Optimization

.

semi-infinite

.

.

.

polytopic uncertainty

Robust Optimization

.

.

.

robust portfolio optimization

Robust Optimization

.

$T

r

.

.

,

solution of robust portfolio optimization problem

Robust Optimization

CQP

,,

.

.

robust CQP

Robust Optimization

.

CQP

,

.

,

Ex.: robust least-squares

Robust Optimization

.

..

References