Global Resolution of Convex Programs with Complementarity Constraints

Abstract

Abstract : This collaborative project aims at the study of the global resolution of convex programs with complementarity constraints (CPCCs), which form a large subclass of the class of mathematical programs with complementarity constraints (MPCCs). Despite the large literature on the local properties of an MPCC, there is a lack of systematic investigation on the computation of a globally optimal solution of these constrained optimization problems, or in the case where such a solution does not exist, on the generation of a certificate demonstrating the solution non-existence. While this is by no means an easy task, the pervasiveness of the CPCC in applications provides an important motivation for the undertaking of this project. As a first step in our study, we have focused on the class of linear programs with linear complementarity constraints (LPCCs), which is essentially a linear program with additional complementarity constraints on certain pairs of variables. Subsequently, we have extended our work to the class of convex quadratic programs with complementarity constraints (QPCCs). Much research remains to be done; we believe that our work so far has provided a solid foundation for the global resolution of this highly challenging class of practically important constrained optimization problems.