A Smoothing Penalized Sample Average Approximation Method for Stochastic Programs with Second Order Stochastic Dominance Constraints

摘要

In this paper, we propose a smoothing penalized sample average approximation (SAA) method for solving a stochastic minimization with second order dominance constraints. The basic idea is to use sample average to approximate the expected values of the underlying random functions and then reformulate the discretised problem as an ordinary nonlinear programming problem with finite number of constraints. An exact penalty function method is proposed to deal with the latter and an elementary smoothing technique is used to tackle the nonsmoothness of the plus function and the exact penalty function. We investigate the convergence of the optimal value obtained from solving the smoothed penalized sample average approximation problem as sample size increases and show that with probability approaching to one at exponential rate with the increase of sample size the optimal value converges to its true counterpart. Some preliminary numerical results are reported.