Nonsmooth Mechanics Meeting: June 14, ferris/talks/aussois-intro.pdf · Nonsmooth Mechanics Meeting:

  • View
    212

  • Download
    0

Embed Size (px)

Text of Nonsmooth Mechanics Meeting: June 14, ferris/talks/aussois-intro.pdf · Nonsmooth Mechanics...

  • An Introduction to Complementarity

    Michael C. Ferris

    University of Wisconsin, Madison

    Nonsmooth Mechanics Meeting: June 14, 2010

    Ferris (Univ. Wisconsi) EMP Aussois, June 2010 1 / 63

  • Outline

    Introduction to Complementarity Models

    Extension to Variational Inequalities

    Extended Mathematical Programming

    Heirarchical Optimization

    Introduction: Transportation Model

    Application: World Dairy Market Model

    Algorithms: Feasible Descent Framework

    Implementation: PATH

    Results

    Ferris (Univ. Wisconsi) EMP Aussois, June 2010 2 / 63

  • Sample Network

    S3

    S2

    S1

    D1

    D2

    D3

    D4

    Ferris (Univ. Wisconsi) EMP Aussois, June 2010 3 / 63

  • Transportation Model

    Suppliers ship good from warehouses to customersI Satisfy demand for commodityI Minimize transportation cost

    Transportation network provided as set A of arcsVariables xi ,j - amount shipped over (i , j) AParameters

    I si - supply at node iI di - demand at node iI ci,j - cost to ship good from nodes i to j

    Ferris (Univ. Wisconsi) EMP Aussois, June 2010 4 / 63

  • Linear Program

    minx0

    (i ,j)A ci ,jxi ,jsubject to

    j :(i ,j)A xi ,j si ii :(i ,j)A xi ,j dj j

    Ferris (Univ. Wisconsi) EMP Aussois, June 2010 5 / 63

  • Multipliers

    Introduce multipliers (marginal prices) ps and pdj :(i ,j)A xi ,j si psi 0

    In a competitive marketplacej :(i ,j)A xi ,j < si psi = 0

    At solution j :(i ,j)A xi ,j = si or p

    si = 0

    Complementarity relationshipj :(i ,j)A xi ,j si psi 0

    Ferris (Univ. Wisconsi) EMP Aussois, June 2010 6 / 63

  • Wardropian Equilibrium

    Delivery cost exceeds market price

    psi + ci ,j pdj

    Strict inequality implies no shipment xi ,j = 0

    Linear complementarity problemj :(i ,j)A xi ,j si psi 0 i

    dj

    i :(i ,j)A xi ,j pdi 0 jpdj psi + ci ,j xi ,j 0 (i , j) A

    First order conditions for linear program

    Ferris (Univ. Wisconsi) EMP Aussois, June 2010 7 / 63

  • Nonlinear Complementarity Problems

    Given F :

  • Extensions

    Original problem has fixed demand

    Use general demand function d(p)

    ExamplesI Linear demand

    i :(i,j)A

    xi,j dj(1 pdj )

    I Nonlinear demandF Cobb-DouglasF CES function

    Use more general cost functions c(x)

    Ferris (Univ. Wisconsi) EMP Aussois, June 2010 9 / 63

  • Taxes and Tariffs

    Exogenous supply tax ti

    psi (1 + ti ) + ci ,j pdj

    Endogenous taxesI Make ti a variableI Add driving equation

    No longer optimality conditions

    Most complementarity problems do not correspond to firstorder conditions of optimization problems

    Ferris (Univ. Wisconsi) EMP Aussois, June 2010 10 / 63

  • Use of complementarity

    Pricing electricity markets and options

    Contact Problems with FrictionVideo games: model contact problems

    I Friction only occurs if bodies are in contact

    Crack PropagationStructure design

    I how springy is concreteI optimal sailboat rig design

    Congestion in NetworksComputer/traffic networks

    I The price of anarchy measures difference between system optimal(MPCC) and individual optimization (MCP)

    Electricity Market Deregulation

    Game Theory (Nash Equilibria)

    General Equilibria with Incomplete Markets

    Impact of Environmental Policy Reform

    Ferris (Univ. Wisconsi) EMP Aussois, June 2010 11 / 63

  • World Dairy Market Model

    Spatial equilibrium model of world dairy sectorI 5 farm milk typesI 8 processed goodsI 21 regions

    Regions trade raw and processed goods

    Barriers to free tradeI Import policies: quotas, tariffsI Export policies: subsidies

    Study impact of policy decisionsI GATT/URAAI Future trade negotiations

    Ferris (Univ. Wisconsi) EMP Aussois, June 2010 12 / 63

  • Formulation

    Quadratic programI Variables: quantitiesI Constraints: production and transportationI Objective: maximize net social welfare

    Difficulty is ad valorem tariffsI Tariff based on value of goodsI Market value is multiplier on constraint

    Complementarity problemI Formulate optimality conditionsI Market price is now a variableI Directly model ad valorem tariffs

    Ferris (Univ. Wisconsi) EMP Aussois, June 2010 13 / 63

  • World Dairy Market Model Statistics

    Quadratic programI 31,772 variablesI 14,118 constraints

    Linear complementarity problemI 45,890 variables and constraintsI 131,831 nonzeros

    Ferris (Univ. Wisconsi) EMP Aussois, June 2010 14 / 63

  • European Put Option

    Contract where holder can sell an asset1 At a fixed expiration time T2 For a fixed price E

    Asset value S(t) satisfies

    dS = ( dX + dt)S

    I dX is random returnI dt is deterministic return

    Black-Scholes Equation

    V

    t+

    1

    22S2

    2V

    S2+ rS

    V

    S rV = 0

    Ferris (Univ. Wisconsi) EMP Aussois, June 2010 15 / 63

  • American Put Options

    Contract where holder can sell an asset1 At any time prior to a fixed expiration T2 For a fixed price E

    Free Boundary Sf (t) optimal exercise price

    V (Sf (t), t) = max(E Sf (t), 0)

    V

    S(Sf (t), t) = 1

    If S Sf (t) then satisfy Black-Scholes Equation

    V

    t+

    1

    22S2

    2V

    S2+ rS

    V

    S rV = 0

    Ferris (Univ. Wisconsi) EMP Aussois, June 2010 16 / 63

  • Complementarity Problem

    Reformulate to remove dependence on boundary

    Partial differential complementarity problem

    F (S , t) Vt +12

    2S2 2V

    S2+ rS VS rV

    0 F (S , t) V (S , t) max(E S , 0)

    Free boundary recovered after solving

    Ferris (Univ. Wisconsi) EMP Aussois, June 2010 17 / 63

  • Solution Method

    Finite differences used to discretizeI Central differences for spaceI Forward differences for timeI Crank-Nicolson method

    Step through time from T to present

    At each t a linear complementarity problem is solved

    Used GAMS/PATH to model and solve

    Ferris (Univ. Wisconsi) EMP Aussois, June 2010 18 / 63

  • Results

    4 6 8 10 12 14 160

    1

    2

    3

    4

    5

    6

    7

    Asset Price (S)

    Opt

    ion

    Val

    ue (

    V)

    payofft = 0

    Ferris (Univ. Wisconsi) EMP Aussois, June 2010 19 / 63

  • Background

    Nonlinear equations F (x) = 0

    Newtons Method

    F (xk) +F (xk)dk = 0xk+1 = xk + dk

    Damp using Armijo linesearch on 12 F (x)22

    Descent direction - gradient of merit function

    PropertiesI Well definedI Global and local-fast convergence

    Ferris (Univ. Wisconsi) EMP Aussois, June 2010 20 / 63

  • Cycling Example

    1.5 1 0.5 0 0.5 1 1.52.5

    2

    1.5

    1

    0.5

    0

    0.5

    1

    1.5

    2

    2.5

    x

    f(x)

    Ferris (Univ. Wisconsi) EMP Aussois, June 2010 21 / 63

  • Merit Function

    1.5 1 0.5 0 0.5 1 1.50

    0.5

    1

    1.5

    2

    2.5

    3

    3.5

    x

    0.5(f(x))2

    Ferris (Univ. Wisconsi) EMP Aussois, June 2010 22 / 63

  • Normal Map

    Equivalent piecewise smooth equation F+(x) = 0

    F+(x) F (x+) + x x+

    Nonsmooth Newton MethodI Piecewise linear system of equationsI Solve via a pivotal methodI Damp using Armijo search on 12 F+(x)

    22

    PropertiesI Global and local-fast convergenceI Merit function not differentiable

    Ferris (Univ. Wisconsi) EMP Aussois, June 2010 23 / 63

  • Nonlinear Complementarity Problem

    Assume F+(x) = 01 If xi 0 then xi (xi )+ = 0 and Fi (x) = 02 If xi 0 then xi (xi )+ 0 and Fi (x) 0

    Therefore x+ solves0 F (x) x 0

    xTF (x) = 0

    Compact representation

    0 F (x) x 0

    If z solves NCP(F ) then F+(z F (z)) = 0

    Ferris (Univ. Wisconsi) EMP Aussois, June 2010 24 / 63

  • Piecewise Linear Example

    0.5 0.4 0.3 0.2 0.1 0 0.1 0.2 0.3 0.4 0.50.5

    0.4

    0.3

    0.2

    0.1

    0

    0.1

    0.2

    0.3

    0.4

    0.5

    x1

    x2

    Ferris (Univ. Wisconsi) EMP Aussois, June 2010 25 / 63

  • Fischer-Burmeister Function

    (x) defined componentwise

    i (x)

    (xi )2 + (Fi (x))2 xi Fi (x)

    (x) = 0 if and only if x solves NCP(F )

    Not continuously differentiable - semismooth

    Natural merit function (12 (x)22) is differentiable

    Ferris (Univ. Wisconsi) EMP Aussois, June 2010 26 / 63

  • Fischer-Burmeister Example

    0.5 0.4 0.3 0.2 0.1 0 0.1 0.2 0.3 0.4 0.50.5

    0.4

    0.3

    0.2

    0.1

    0

    0.1

    0.2

    0.3

    0.4

    0.5

    x1

    x2

    Ferris (Univ. Wisconsi) EMP Aussois, June 2010 27 / 63

  • Review

    Nonlinear Complementarity Problem

    Piecewise smooth system of equationsI Use nonsmooth Newton MethodI Solve linear complementarity problem per iterationI Merit function not differentiable

    Fischer-BurmeisterI Differentiable merit function

    Combine to obtain new algorithmI Well definedI Global and local-fast convergence

    Ferris (Univ. Wisconsi) EMP Aussois, June 2010 28 / 63

  • Feasible Descent Framework

    Calculate direction using a local methodI Generates feasible iteratesI Local fast convergenceI Used nonsmooth Newton Method

    Accept direction if descent for 12 (x)2

    Otherwise use projected gradient step

    Ferris (Univ. Wisconsi) EMP Aussois, June 2010 29 / 63

  • Theorem

    Let {xk}

  • Computational Details

    Crashing method to quickly identify basis

    Nonmonotone search with watchdog

    Per