! Mathematical Programs with Equilibrium Constraints and Applications to Control

We discuss recent advances in mathematical programs with equilibrium constraints (MPECs). We describe the challenges posed by these problems and the current algorithmic solutions. We emphasize in particular the use of the elastic mode approach. We also present initial investigations in applications of MPECs to control problems. Ceywords: Complementarity Constraints, Equilibrium Constraints, Elastic Mode, Nonlinear Programming. 1. MPEC: APPLICATIONS AND CURRENT APPROACHES The paper presents a summarizes of recent work on mathematical programs with equilibrium constraints (MPECs). When the equilibrium constraints occur from optimality conditions over polyhedral cones, the problem becomes a mathematical program with complementarity constraints (MPCC). In this work we will use the term MPEC to describe MPCC as well, since this term is far more popular in the current complementarity literature. 1.1 Formulation and difficulty At the core of MPECs is the complementarity constraint. We say that variables are complementary, or that they satisfy a complementarity constraint, if they satisfy the relationship We denote this relationship between the variables by . Note that we can write the same relationship between vector components, provided that we use the scalar product to define the relevant product, that is, . , a b 0, 0, 0. a b ab ! ! " a b # 0 T ab a b " " An MPEC is a nonlinear program that contains complementarity constraints between suitable variables, in addition to other type of constraints: