Obtaining Tighter Relaxations of Mathematical Programs with Complementarity Constraints

The class of mathematical programs with complementarity constraints (MPCCs) constitutes a powerful modeling paradigm. In an effort to find a global optimum, it is often useful to examine the relaxation obtained by omitting the complementarity constraints. We discuss various methods to tighten the relaxation by exploiting complementarity, with the aim of constructing better approximations to the convex hull of the set of feasible solutions to the MPCC, and hence better lower bounds on the optimal value of the MPCC. Better lower bounds can be useful in branching schemes to find a globally optimal solution. Different types of linear constraints are constructed, including cuts based on bounds on the variables and various types of disjunctive cuts. Novel convex quadratic constraints are introduced, with a derivation that is particularly useful when the number of design variables is not too large. A lifting process is specialized to MPCCs. Semidefinite programming constraints are also discussed. All these constraints are typically applicable to any convex program with complementarity constraints. Computational results for linear programs with complementarity constraints (LPCCs) are included, comparing the benefit of the various constraints on the value of the relaxation, and showing that the constraints can dramatically speed up the solution of the LPCC.