A Benders decomposition method for discretely-constrained mathematical programs with equilibrium constraints

We present a new methodology to solve discretely-constrained mathematical programs with equilibrium constraints (DC-MPECs). Typically these problems include an upper planning-level optimization with some discrete decision variables (eg, build/don’t build) as well as a lower operations-level problem often described by an optimization or nonlinear complementarity problem. This lower-level problem may also include some discrete variables. MPECs are very challenging problems to solve and the inclusion of integrality constraints makes this class of problems even more computationally difficult. We develop a new variant of the Benders algorithm combined with a heuristic procedure that decomposes the domain of the upper-level discrete variables to solve the resulting DC-MPECs. We provide convergence theory as well as a number of numerical examples, some derived from energy applications, to validate the new method. It should be noted that the convergence theory applies if the heuristic procedure correctly identifies a decomposition of the domain so that the lower-level problem's optimal value function is convex. This is challenging but our numerical results are positive.