On the Robustness of Global Optima and Stationary Solutions to Stochastic Mathematical Programs with Equilibrium Constraints, Part 2: Applications

Abstract

We consider a stochastic mathematical program with equilibrium constraints (SMPEC), and show that under certain assumptions, global optima and stationary solutions are robust with respect to changes in the underlying probability distribution. In particular, the discretization scheme Sample Average Approximation, which is convergent for both global optima and stationary solutions, can be combined with the robustness result to motivate the use of SMPECs in practice.We also consider SMPECs with multiple objectives, and establish robustness of weakly Pareto optimal and stationary solutions.Two applications are presented, both principally and numerically, in order to exemplify the use of SMPECs: a classic traffic network design problem where travel costs are uncertain, and the optimization of a treatment plan in intensity modulated radiation therapy, where radiobiological parameters are uncertain.