Mathematical programming with equilibrium constraints
Author: Alexandra Rodriguez (ChE 345 Spring 2015)
Stewards: Dajun Yue and Fengqi You
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:
Therefore, an equilibrium constrained optimization model is given by:
MPEC plays a central role in the modeling of transportation problems, economics, and engineering design.
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.
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):
The complementarity constraints can be written equivalently as:
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:
 G.B. Allende. Mathematical programs with equilibrium constraints: solution techniques from parametric optimization (1977).
 M.C. Ferris, S.P. Dirkse, A. Meeraus. Mathematical programs with equilibrium constraints: automatic reformation and solution via constrained optimization. Northwestern University (2002).
 H. Pieper. Algorithms for mathematical programs with equilibrium constraints with applications to deregulated electricity markets. Stanford University (2001).
 R. Andreani, J.M. Martinez. On the solution of mathematical programming problems with equilibrium constraints (2008).