Combined Monte Carlo sampling and penalty method for Stochastic nonlinear complementarity problems

Abstract

Abstract. In this paper, we consider a new formulation with recourse for a class of stochastic nonlinear complementarity problems. We show that the new formulation is equivalent to a smooth semi-infinite program that no longer contains recourse variables. We then propose a combined Monte Carlo sampling and penalty method for solving the problem in which the underlying sample space is assumed to be compact. Furthermore, we suggest a compact approximation approach for the case where the sample space is unbounded. Two preliminary numerical examples are included as well.

Linear Complementarity Problems on Extended Second Order Cones

“…(iii) Stochastic mathematical programs with equilibrium constraints (SMPEC) reformulation, introduced by Lin and Fukushima [4,10,11]. This method highlights a recourse variate z(ω) to compensate the violation of complementarity in (4) for some outcomes of ω ∈ Ω, then it reformulates (4) to the following model:…”

“…If partial or all of the coefficients in the LCP are uncertain, the LCP will be turned into a stochastic linear complementarity problem (SLCP), which is firstly introduced by Chen and Fukushima [1]. Articles about SLCP can be found in [2][3][4][5].…”

The Distributionally Robust Optimization Reformulation for Stochastic Complementarity Problems

“…In this section, we give preliminary numerical results with an example of a linear complementarity problem with a random parameter from [8] for the proposed method.…”

“…In this way, people seek some deterministic and approximate reformulations of the SCP (1). Among the literature of the SCP, there are some typical deterministic reformulations, such as, the expected value method, the expected residual minimization method, stochastic mathematical program with equilibrium constraints reformulation, and the CVaR minimization reformulation (see [5][6][7][8] for details). Because the sample average approximation method is utilized to solve the deterministic reformulations of the SCP, the reformulations of the SCP are depended on the sample of the random data which itself has the property of the randomness or uncertainty.…”