A Sample Average Approximation Regularization Method For A Stochastic Mathematical Program With General Vertical Complementarity Constraints

Stochastic mathematical program with general vertical complementarity constraints(SMPVCC)is an extension form of the deterministic mathematical program with general vertical complementarity constraints(MPVCC),which contains stochastic mathematical program with complementarity constraints(SMPCC)as its special case and have wide applications in engineering mechanics,transportation,finance and economics,and other fields.The study of theory of SMPVCC problem(such as stationary conditions and constraint qualification,etc.)have relatively perfect,but the research on the algorithm need further developing,so the research of the algorithm for solving SMPVCC have an important significance.Based on the log-exponential function,a sample average approximation(SAA)regularization method is proposed in this paper for solving SMPVCC,and the detailed analysis of the convergence of this method has carried out,the main research contents are as follows:First,under the regular condition,the relationship between the optimal solutions of regularized sample average approximation problem and the ones of original problem has established,namely when the parameter tends to zero and the sample size tends to infinity,any accumulation point of the optimal solution sequence of regularized sample average approximation problem with probability 1 is an optimal solution of the SMPVCC problem.Second,under the generalized Mangasarian-Fromovitz constraint qualification,the relationship between the sequence of optimal value of regularized sample average approximate SMPVCC and the optimal value of the original problem is established,namely the sequence of optimal value of regularized sample average approximate SMPVCC converges to the optimal value of the original problem with probability 1 with exponential convergence speed.At the same time the consistency between stationary point of regularized sample average approximation problem and the SMPVCC is proved.Finally,based on the fact that the Stackelberg game theory can be described as a SMPVCC problem,we apply the sample average approximation regularization method to this problem and obtain an equilibrium solution.