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A probability metrics approach for reducing the bias of optimality gap estimators in two-stage stochastic linear programming

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Abstract

Monte Carlo sampling-based estimators of optimality gaps for stochastic programs are known to be biased. When bias is a prominent factor, estimates of optimality gaps tend to be large on average even for high-quality solutions. This diminishes our ability to recognize high-quality solutions. In this paper, we present a method for reducing the bias of the optimality gap estimators for two-stage stochastic linear programs with recourse via a probability metrics approach, motivated by stability results in stochastic programming. We apply this method to the Averaged Two-Replication Procedure (A2RP) by partitioning the observations in an effort to reduce bias, which can be done in polynomial time in sample size. We call the resulting procedure the Averaged Two-Replication Procedure with Bias Reduction (A2RP-B). We provide conditions under which A2RP-B produces strongly consistent point estimators and an asymptotically valid confidence interval. We illustrate the effectiveness of our approach analytically on a newsvendor problem and test the small-sample behavior of A2RP-B on a number of two-stage stochastic linear programs from the literature. Our computational results indicate that the procedure effectively reduces bias. We also observe variance reduction in certain circumstances.

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Authors and Affiliations

  1. Graduate Interdisciplinary Program in Applied Mathematics, University of Arizona, Tucson, AZ, 85721, USA

    Rebecca Stockbridge

  2. Department of Systems and Industrial Engineering, University of Arizona, Tucson, AZ, 85721, USA

    Güzin Bayraksan

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

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Stockbridge, R., Bayraksan, G. A probability metrics approach for reducing the bias of optimality gap estimators in two-stage stochastic linear programming. Math. Program. 142, 107–131 (2013). https://doi.org/10.1007/s10107-012-0563-6

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