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Variance reduction in Monte Carlo sampling-based optimality gap estimators for two-stage stochastic linear programming

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Abstract

This paper presents a comparative computational study of the variance reduction techniques antithetic variates and Latin hypercube sampling when used for assessing solution quality in stochastic programming. Three Monte Carlo sampling-based procedures that provide point and interval estimators of optimality gap are considered: one that uses multiple replications, and two others with an alternative sample variance estimator that use single or two replications. Theoretical justification for using these alternative sampling techniques is discussed. In particular, we discuss asymptotic properties of the resulting estimators using Latin hypercube sampling for single- and two-replication procedures in detail. These theoretical considerations result in some subtle changes in the implementation of the procedures. A collection of two-stage stochastic linear test problems with different characteristics is used to empirically compare the three procedures for assessing solution quality with these variance reduction techniques.

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Acknowledgments

The authors are grateful to Tito Homem-de-Mello for providing reference [11], the coordinating editor and the referees for suggestions that improved the paper, and Andrzej Ruszczyński and Artur Świetanowski for access to their regularized decomposition code, which was used to solve the test problems. In addition, allocations of computer time from the Ohio Supercomputer Center (http://osc.edu/ark:/19495/f5s1ph73), UA Research Computing High Performance Computing (HPC) and High Throughput Computing (HTC) at the University of Arizona, and the Wayne State University High Performance Computing Services are gratefully acknowledged. This research has been partially funded by the National Science Foundation Grant CMMI-1345626.

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Authors and Affiliations

  1. General Motors, Warren, MI, 48092, USA

    Rebecca Stockbridge

  2. Department of Integrated Systems Engineering, The Ohio State University, Columbus, OH, 43210, USA

    Güzin Bayraksan

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  1. Rebecca Stockbridge
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  2. Güzin Bayraksan
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Correspondence to Rebecca Stockbridge.

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Stockbridge, R., Bayraksan, G. Variance reduction in Monte Carlo sampling-based optimality gap estimators for two-stage stochastic linear programming. Comput Optim Appl 64, 407–431 (2016). https://doi.org/10.1007/s10589-015-9814-9

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