# Difference between revisions of "Mathematical programming with equilibrium constraints"

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<math> min </math> <math> c^T (x,y) </math> <br\> | <math> min </math> <math> c^T (x,y) </math> <br\> | ||

− | <math> s.t. </math> <math> C(x,y) + d - BY^T </math> & | + | <math> s.t. </math> <math> C(x,y) </math><math>+d</math><math>-BY^T </math> λ <math> = 0 </math> <br\> |

<math> B(x,y) \ge b </math> <br\> | <math> B(x,y) \ge b </math> <br\> | ||

− | | + | λ <math> \ge 0 </math> <br\> |

− | <math> [B(x,y) - b]^T = 0 </math> <br\> | + | <math> [B(x,y) </math><math> - b]^T = 0 </math> <br\> |

− | <math> where b, d, B, C \in \mathbb{R} </math><br\> | + | <math> where </math><math> b, </math><math>d, </math><math>B, </math><math>C \in \mathbb{R} </math><br\> |

=Conclusion= | =Conclusion= |

## Revision as of 21:30, 24 May 2015

## 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

# Conclusion

# 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).