Simulation-based confidence bounds for two-stage stochastic programs

• Published in: Mathematical programming Vol. 138; no. 1-2; pp. 15 - 42
• Main Authors: Glynn, Peter W., Infanger, Gerd
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer-Verlag 01.04.2013; Springer Nature B.V
• Subjects: Algorithms; Approximation; Asymptotic properties; Calculus of Variations and Optimal Control; Optimization; Combinatorics; Confidence; Confidence intervals; Decomposition; Full Length Paper; Iterative methods; Linear programming; Mathematical and Computational Physics; Mathematical Methods in Physics; Mathematics; Mathematics and Statistics; Mathematics of Computing; Numerical Analysis; Random number sampling; Reduction; Sample size; Sampling; Stochasticity; Theoretical; 90C15 Stochastic programming; 97K50 Probability theory; 90C05 Linear programming; 62G15 Tolerance and confidence regions
• ISSN: 0025-5610; 1436-4646
• DOI: 10.1007/s10107-012-0621-0

This paper provides a rigorous asymptotic analysis and justification of upper and lower confidence bounds proposed by Dantzig and Infanger (A probabilistic lower bound for two-stage stochastic programs, Stanford University, CA, 1995 ) for an iterative sampling-based decomposition algorithm, introduced by Dantzig and Glynn (Ann. Oper. Res. 22:1–21, 1990 ) and Infanger (Ann. Oper. Res. 39:41–67, 1992 ), for solving two-stage stochastic programs. The paper provides confidence bounds in the presence of both independent sampling across iterations, and when common samples are used across different iterations. Confidence bounds for sample-average approximation then follow as a special case. Extensions of the theory to cover use of variance reduction and the dropping of cuts are also presented. An extensive empirical investigation of the performance of these bounds establishes that the bounds perform reasonably on realistic problems.