Bias and Variance Reduction in Assessing Solution Quality for Stochastic Programs

Stochastic programming combines ideas from deterministic optimization with probability and statistics to produce more accurate models of optimization problems involving uncertainty. However, due to their size, stochastic programming problems can be extremely difficult to solve and instead approximate solutions are used. Therefore, there is a need for methods that can accurately identify optimal or near optimal solutions. In this dissertation, we focus on improving Monte-Carlo sampling-based methods that assess the quality of potential solutions to stochastic programs by estimating optimality gaps. In particular, we aim to reduce the bias and/or variance of these estimators. We first propose a technique to reduce the bias of optimality gap estimators which is based on probability metrics and stability results in stochastic programming. This method, which requires the solution of a minimum-weight perfect matching problem, can be run in polynomial time in sample size. We establish asymptotic properties and present computational results. We then investigate the use of sampling schemes to reduce the variance of optimality gap estimators, and in particular focus on antithetic variates and Latin hypercube sampling. We also combine these methods with the bias reduction technique discussed above. Asymptotic properties of the resultant estimators are presented, and computational results on a range of test problems are discussed. Finally, we apply methods of assessing solution quality using antithetic variates and Latin hypercube sampling to a sequential sampling procedure to solve stochastic programs. In this setting, we use Latin hypercube sampling when generating a sequence of candidate solutions that is input to the procedure. We prove that these procedures produce a high-quality solution with high probability, asymptotically, and terminate in a finite number of iterations. Computational results are presented