# Mathematical programming with equilibrium constraints

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:

φ

Therefore, an equilibrium constrained optimization model is given by:

φ

φ

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

# Problem formulation

φ

φ

## 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):

The complementarity constraints can be written equivalently as:

perp-to

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:

λ

λ

## Applications

As mentioned above, there are several applications of MPEC problems. Two published examples deal with economics and mathematical physics.

### Economics

For this application, consider that *n* companies product the same product. We will introduce an integer variable *y* to denote the number of units a company will sell. We will further denote *y _{i}* as the number of items that company

*i*decides to sell. The total price of the product on the market will be notated P(T), where T = . The total cost of production for a company will be given by

*f*. With this notation, the profit can then be given by the expression

_{i}(y_{i})# 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).