# Mathematical programming with equilibrium constraints

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

# 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 $\ge 0$ $[B(x,y)$ $- b]^T = 0$ $where$ $b,$ $d,$ $B,$ $C \in \mathbb{R}$