Stochastic mathematical programs with equilibrium constraints (SMPEC) is a framework of optimization problems in which the objective function in terms of the main (“upper-level”) variables is evaluated through the solution of a secondary optimization problem (or variational inequality), whose data is uncertain. The goal is to find a good compromise “design” such that this “response” is favourable for the upper-level objective. As the response is stochastic the evaluation of the design must be based on a type of average value over the range of responses, such as a risk-based objective like CVaR (conditional value-at-risk). The project has developed several theoretical results concerning the robustness of such designs with respect to changes in the underlying probability distribution, as well as convergence results for discretization algorithms, even for the case of multiple upper-level objectives. Applications studied include robust link tolls in traffic networks, and robust intensity modulated radiation therapy.
Keywords: Stochastic mathematical programs with equilibrium constraints, sample average approximation, conditional value-at-risk, Pareto optimal stationary solutions, robustness of solutions
Gothenburg Mathematical Modelling Centre, Chalmers Area of Advance Transport