Solving stochastic mathematical programs with complementarity constraints using simulation

We consider stochastic mathematical programs with complementarity constraints in which both the objective and constraints involve limit functions that need to be approximated. Such programs can be used for modeling ‘average’ (expected) or steady–state behavior of complex stochastic systems. We first describe these stochastic mathematical programs with complementarity constraints and compare them with different stochastic mathematical programs with equilibrium constraints from the literature. This explicit discussion may facilitate selecting an appropriate stochastic model. We then describe a simulation–based method called sample–path optimization for solving these problems and provide sufficient conditions under which appropriate approximating problems will have solutions converging to a solution of the original problem almost surely. We illustrate an application on toll pricing in transportation networks. We explain how uncertainty can be incorporated and the approximating problems are solved using an off–the–shelf solver. These developments enable solving certain stochastic bilevel optimization problems and Stackelberg games using simulation.