On sample average approximation for two-stage stochastic programs without relatively complete recourse

We investigate sample average approximation (SAA) for two-stage stochastic programs without relatively complete recourse, i.e., for problems in which there are first-stage feasible solutions that are not guaranteed to have a feasible recourse action. As a feasibility measure of the SAA solution, we consider the recourse likelihood, which is the probability that the solution has a feasible recourse action. For epsilon is an element of (0, 1), we demonstrate that the probability that a SAA solution has recourse likelihood below 1 - epsilon converges to zero exponentially fast with the sample size. Next, we analyze the rate of convergence of optimal solutions of the SAA to optimal solutions of the true problem for problems with a finite feasible region, such as bounded integer programming problems. For problems with non-finite feasible region, we propose modified padded SAA problems and demonstrate in two cases that such problems can yield, with high confidence, solutions that are certain to have a feasible recourse decision. Finally, we conduct a numerical study on a two-stage resource planning problem that illustrates the results, and also suggests there may be room for improvement in some of the theoretical analysis.

Automated summary

The study shows that the feasibility likelihood of SAA solutions converges exponentially fast to zero with the sample size, especially for problems with a finite feasible region. For problems with a non-finite feasible region, modified padded SAA problems are proposed to generate solutions with a feasible recourse decision with high confidence.