Combined Smoothing Implicit Programming and Penalty Method for Stochastic Mathematical Programs with Equilibrium Constraints

In this paper, we consider the stochastic mathematical program with equilibrium constraints (SMPEC), which can be thought as a generalization of the mathematical program with equilibrium constraints. Many decision problems can be formulated as SMPECs in practice. We discuss both here-and-now and lower-level wait-and-see decision problems. In particular, with the help of a penalty technique, we propose a combined smoothing implicit programming and penalty method for the here-and-now decision problem and a comprehensive convergence theory is also included. Furthermore, we remark that similar discussion applies to the lower-level wait-and-see model as well.