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

TL;DR

This paper analyzes the sample average approximation method for two-stage stochastic programs lacking guaranteed feasible recourse, establishing exponential convergence of recourse likelihood and proposing modifications for non-finite feasible regions.

Contribution

It introduces a new feasibility measure, the recourse likelihood, and provides convergence analysis and modifications for problems without relatively complete recourse.

Findings

Recourse likelihood converges exponentially fast with sample size.

Modified 'padded' SAA problems can ensure feasible recourse with high confidence.

Numerical results support theoretical findings and highlight potential improvements.

Abstract

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 $\epsilon \in (0,1)$, 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.