Difference between revisions of "Mathematical programming with equilibrium constraints"

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  &nbsp;  &nbsp; &nbsp; <math> [B(x,y) </math><math> - b]^T = 0 </math> <br\>
 
  &nbsp;  &nbsp; &nbsp; <math> [B(x,y) </math><math> - b]^T = 0 </math> <br\>
 
  &nbsp; <math> where </math> &nbsp; <math> b, </math><math>d, </math><math>B, </math><math>C \in \mathbb{R} </math><br\>
 
  &nbsp; <math> where </math> &nbsp; <math> b, </math><math>d, </math><math>B, </math><math>C \in \mathbb{R} </math><br\>
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=References=
 
=References=

Revision as of 22:41, 24 May 2015

Author: Alexandra Rodriguez (ChE 345 Spring 2015)
Stewards: Dajun Yue and Fengqi You


Contents

Introduction

Mathematical programming with equilibrium constraints (MPEC) is a type of nonlinear programming with constrained optimization. Constraints must satisfy an equilibrium condition, of which the simplest form is given by the critical point:

\nabla_y φ = 0

Therefore, an equilibrium constrained optimization model is given by: 

  min f(x,y)

  s.t. \nabla_y  φ  (x,y) = 0 

  where f,  φ  \in \mathbb{R}

MPEC plays a central role in the modeling of transportation problems, economics, and engineering design.

Problem formulation

  min     f(x,y)

   s.t.      g(x,y) \ge 0 

   y \in Y(x) 

  φ(x,y,z) \ge 0 

   where      f, g,  φ, Y(x) \in \mathbb{R}

Feasible set

For a feasible set, the conditions for convexity and closedness are as follows. If Y(x) is convex, and functions f, g, and ∅ are concave, then the mathematical program is convex. Furthermore, if the Mangasarian-Fromovitz constraint qualification holds at all z ∈ Y(x), then Y(x) is the lower semi-continuous bound, and the mathematical program is closed.

KKT transformation

Complementarity constrained optimization

By applying the Karush-Kuhn-Tucker (KKT) approach to solving an equilibrium constraint problem (EC), a program with complementarity constraints can be obtained (CC): 

   min     f(x)  

   s.t.      g(x) \ge 0, 

   G_1(x)    \ge 0, 

   G_2(x)   \ge 0, 

   G_1(x)   G_2(x)   = 0

The complementarity constraints can be written equivalently as:

0 \le G_1(x) perp-to G_2(x) \ge 0.

A mathematical program with complementarity constraints (MPCC) is a relaxed MPEC.

Linear constrained optimization

The KKT approach may also lead to an MPCC with only linear functions: 

   min     c^T (x,y)  

   s.t.       C(x,y) +d-BY^T  λ  = 0  

         B(x,y) \ge b  

         λ  \ge 0  

         [B(x,y)  - b]^T = 0  

   where     b, d, B, C \in \mathbb{R}

References

[1] G.B. Allende. Mathematical programs with equilibrium constraints: solution techniques from parametric optimization (1977).
[2] M.C. Ferris, S.P. Dirkse, A. Meeraus. Mathematical programs with equilibrium constraints: automatic reformation and solution via constrained optimization. Northwestern University (2002).
[3] H. Pieper. Algorithms for mathematical programs with equilibrium constraints with applications to deregulated electricity markets. Stanford University (2001).
[4] R. Andreani, J.M. Martinez. On the solution of mathematical programming problems with equilibrium constraints (2008).

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