Convergence of Stationary Points of Sample Average Two-Stage Stochastic Programs: A Generalized Equation Approach

This paper presents an asymptotic analysis of a Monte Carlo method, variously known as sample average approximation (SAA) or sample path optimization (SPO), for a general two-stage stochastic minimization problem. We study the case when the second-stage problem may have multiple local optima or stationary points that are not global solutions and SAA is implemented using a general nonlinear programming solver that is only guaranteed to find stationary points. New optimality conditions are developed for both the true problem and its SAA problem to accommodate Karush-Kuhn-Tucker points. Because the optimality conditions are essentially stochastic generalized equations, the asymptotic analysis is carried out for the generalized equations first and then applied to optimality conditions. For this purpose, we analyze piecewise continuous (PC) stochastic mappings to understand when their expectations are piecewise continuous and thereby derive exponential convergence of SAA. It is shown under moderate conditions that, with probability one, an accumulation point of the SAA stationary points satisfies a relaxed stationary condition for the true problem and further that, with probability approaching one exponentially fast with increasing sample size, a stationary point of SAA converges to the set of relaxed stationary points. These results strengthen or complement existing results where the second-stage problem is often assumed to have a unique solution and the exponential convergence is focused on how fast a solution of the true problem becomes an approximate solution of an SAA problem rather than the other way around.