An Introduction to Complementarity Michael C. Ferris University of - PowerPoint PPT Presentation

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 D1 S1 D2 S2 D3 S3 D4 Ferris (Univ. Wisconsi) EMP Aussois, June 2010 3 / 63

Transportation Model Suppliers ship good from warehouses to customers ◮ Satisfy demand for commodity ◮ Minimize transportation cost Transportation network provided as set A of arcs Variables x i , j - amount shipped over ( i , j ) ∈ A Parameters ◮ s i - supply at node i ◮ d i - demand at node i ◮ c i , j - cost to ship good from nodes i to j Ferris (Univ. Wisconsi) EMP Aussois, June 2010 4 / 63

Linear Program � min x ≥ 0 ( i , j ) ∈A c i , j x i , j subject to � j :( i , j ) ∈A x i , j ≤ s i ∀ i � i :( i , j ) ∈A x i , j ≥ d j ∀ j Ferris (Univ. Wisconsi) EMP Aussois, June 2010 5 / 63

Multipliers Introduce multipliers (marginal prices) p s and p d p s � j :( i , j ) ∈A x i , j ≤ s i i ≥ 0 In a competitive marketplace p s � j :( i , j ) ∈A x i , j < s i ⇒ i = 0 At solution p s � j :( i , j ) ∈A x i , j = s i or i = 0 Complementarity relationship p s � j :( i , j ) ∈A x i , j ≤ s i ⊥ i ≥ 0 Ferris (Univ. Wisconsi) EMP Aussois, June 2010 6 / 63

Wardropian Equilibrium Delivery cost exceeds market price p s i + c i , j ≥ p d j Strict inequality implies no shipment x i , j = 0 Linear complementarity problem p s � j :( i , j ) ∈A x i , j ≤ s i ⊥ i ≥ 0 ∀ i p d d j ≤ � i :( i , j ) ∈A x i , j ⊥ i ≥ 0 ∀ j p d j ≤ p s i + c i , j ⊥ x i , 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 : ℜ n → ℜ n Find x ∈ ℜ n such that 0 ≤ F ( x ) x ≥ 0 x T F ( x ) = 0 Compactly written 0 ≤ F ( x ) ⊥ x ≥ 0 Equivalent to nonsmooth equation: min( x , F ( x )) = 0 Ferris (Univ. Wisconsi) EMP Aussois, June 2010 8 / 63

Extensions Original problem has fixed demand Use general demand function d ( p ) Examples ◮ Linear demand � x i , j ≥ d j (1 − p d j ) i :( i , j ) ∈A ◮ Nonlinear demand ⋆ Cobb-Douglas ⋆ CES function Use more general cost functions c ( x ) Ferris (Univ. Wisconsi) EMP Aussois, June 2010 9 / 63

Taxes and Tariffs Exogenous supply tax t i p s i (1 + t i ) + c i , j ≥ p d j Endogenous taxes ◮ Make t i a variable ◮ Add driving equation No longer optimality conditions Most complementarity problems do not correspond to first order conditions of optimization problems Ferris (Univ. Wisconsi) EMP Aussois, June 2010 10 / 63

Use of complementarity Pricing electricity markets and options Contact Problems with Friction Video games: model contact problems ◮ Friction only occurs if bodies are in contact Crack Propagation Structure design ◮ how springy is concrete ◮ optimal sailboat rig design Congestion in Networks Computer/traffic networks ◮ 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 sector ◮ 5 farm milk types ◮ 8 processed goods ◮ 21 regions Regions trade raw and processed goods Barriers to free trade ◮ Import policies: quotas, tariffs ◮ Export policies: subsidies • Study impact of policy decisions ◮ GATT/URAA ◮ Future trade negotiations Ferris (Univ. Wisconsi) EMP Aussois, June 2010 12 / 63

Formulation Quadratic program ◮ Variables: quantities ◮ Constraints: production and transportation ◮ Objective: maximize net social welfare • Difficulty is ad valorem tariffs ◮ Tariff based on value of goods ◮ Market value is multiplier on constraint • Complementarity problem ◮ Formulate optimality conditions ◮ Market price is now a variable ◮ Directly model ad valorem tariffs Ferris (Univ. Wisconsi) EMP Aussois, June 2010 13 / 63

World Dairy Market Model Statistics Quadratic program ◮ 31,772 variables ◮ 14,118 constraints Linear complementarity problem ◮ 45,890 variables and constraints ◮ 131,831 nonzeros Ferris (Univ. Wisconsi) EMP Aussois, June 2010 14 / 63

European Put Option Contract where holder can sell an asset At a fixed expiration time T 1 For a fixed price E 2 Asset value S ( t ) satisfies dS = ( σ dX + µ dt ) S ◮ σ dX is random return ◮ µ dt is deterministic return Black-Scholes Equation 2 σ 2 S 2 ∂ 2 V ∂ V ∂ t + 1 ∂ S 2 + rS ∂ V ∂ S − rV = 0 Ferris (Univ. Wisconsi) EMP Aussois, June 2010 15 / 63

American Put Options Contract where holder can sell an asset At any time prior to a fixed expiration T 1 For a fixed price E 2 Free Boundary S f ( t ) – optimal exercise price V ( S f ( t ) , t ) = max( E − S f ( t ) , 0) ∂ V ∂ S ( S f ( t ) , t ) = − 1 If S ≥ S f ( t ) then satisfy Black-Scholes Equation 2 σ 2 S 2 ∂ 2 V ∂ V ∂ t + 1 ∂ S 2 + 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 2 σ 2 S 2 ∂ 2 V ∂ V ∂ t + 1 ∂ S 2 + rS ∂ V F ( S , t ) ≡ ∂ S − 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 discretize ◮ Central differences for space ◮ Forward differences for time ◮ 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 7 payoff t = 0 6 5 Option Value (V) 4 3 2 1 0 4 6 8 10 12 14 16 Asset Price (S) Ferris (Univ. Wisconsi) EMP Aussois, June 2010 19 / 63

Background Nonlinear equations F ( x ) = 0 Newton’s Method F ( x k ) + ∇ F ( x k ) d k = 0 x k +1 = x k + d k 2 � F ( x ) � 2 Damp using Armijo linesearch on 1 2 Descent direction - gradient of merit function Properties ◮ Well defined ◮ Global and local-fast convergence Ferris (Univ. Wisconsi) EMP Aussois, June 2010 20 / 63

Cycling Example 2.5 2 1.5 1 0.5 f(x) 0 − 0.5 − 1 − 1.5 − 2 − 2.5 − 1.5 − 1 − 0.5 0 0.5 1 1.5 x Ferris (Univ. Wisconsi) EMP Aussois, June 2010 21 / 63

Merit Function 3.5 3 2.5 2 0.5(f(x)) 2 1.5 1 0.5 0 − 1.5 − 1 − 0.5 0 0.5 1 1.5 x 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 Method ◮ Piecewise linear system of equations ◮ Solve via a pivotal method 2 � F + ( x ) � 2 ◮ Damp using Armijo search on 1 2 Properties ◮ Global and local-fast convergence ◮ Merit function not differentiable Ferris (Univ. Wisconsi) EMP Aussois, June 2010 23 / 63

Nonlinear Complementarity Problem Assume F + (¯ x ) = 0 If ¯ x i ≥ 0 then ¯ x i − (¯ x i ) + = 0 and F i (¯ x ) = 0 1 If ¯ x i ≤ 0 then ¯ x i − (¯ x i ) + ≤ 0 and F i (¯ x ) ≥ 0 2 Therefore ¯ x + solves 0 ≤ F ( x ) x ≥ 0 x T F ( 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 x2 0 − 0.1 − 0.2 − 0.3 − 0.4 − 0.5 − 0.5 − 0.4 − 0.3 − 0.2 − 0.1 0 0.1 0.2 0.3 0.4 0.5 x1 Ferris (Univ. Wisconsi) EMP Aussois, June 2010 25 / 63

Fischer-Burmeister Function Φ( x ) defined componentwise � ( x i ) 2 + ( F i ( x )) 2 − x i − F i ( x ) Φ i ( x ) ≡ Φ( x ) = 0 if and only if x solves NCP( F ) Not continuously differentiable - semismooth Natural merit function ( 1 2 � Φ( x ) � 2 2 ) is differentiable Ferris (Univ. Wisconsi) EMP Aussois, June 2010 26 / 63

Fischer-Burmeister Example 0.5 0.4 0.3 0.2 0.1 x2 0 − 0.1 − 0.2 − 0.3 − 0.4 − 0.5 − 0.5 − 0.4 − 0.3 − 0.2 − 0.1 0 0.1 0.2 0.3 0.4 0.5 x1 Ferris (Univ. Wisconsi) EMP Aussois, June 2010 27 / 63

Review Nonlinear Complementarity Problem Piecewise smooth system of equations ◮ Use nonsmooth Newton Method ◮ Solve linear complementarity problem per iteration ◮ Merit function not differentiable Fischer-Burmeister ◮ Differentiable merit function Combine to obtain new algorithm ◮ Well defined ◮ Global and local-fast convergence Ferris (Univ. Wisconsi) EMP Aussois, June 2010 28 / 63