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Handbook of Simulation Optimization pp 245–276Cite as

Stochastic Constraints and Variance Reduction Techniques

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Part of the book series: International Series in Operations Research & Management Science ((ISOR,volume 216))

Abstract

We provide an overview of two select topics in Monte Carlo simulation-based methods for stochastic optimization: problems with stochastic constraints and variance reduction techniques. While Monte Carlo simulation-based methods have been successfully used for stochastic optimization problems with deterministic constraints, there is a growing body of work on its use for problems with stochastic constraints. The presence of stochastic constraints brings new challenges in ensuring and testing optimality, allocating sample sizes, etc., especially due to difficulties in determining feasibility. We review results for general stochastic constraints and also discuss special cases such as probabilistic and stochastic dominance constraints. Next, we review the use of variance reduction techniques (VRT) in a stochastic optimization setting. While this is a well-studied topic in statistics and simulation, the use of VRT in stochastic optimization requires a more thorough analysis. We discuss asymptotic properties of the resulting approximations and their use within Monte Carlo simulation-based solution methods.

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Notes

  1. 1.

    Throughout this chapter, we will use the terminology “sample [of size N] from the distribution of \(\xi\)” to indicate a set of N random variables with the same distribution as \(\xi\).

  2. 2.

    Such an assumption is made just for simplicity of notation; the general case can be dealt with via Radon–Nikodym derivatives, see for instance [3].

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Acknowledgements

This work is supported in part by the National Science Foundation under Grant CMMI-1345626, and by Conicyt-Chile under grant Fondecyt 1120244.

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Authors and Affiliations

  1. Universidad Adolfo Ibañez, Santiago, Chile

    Tito Homem-de-Mello

  2. The Ohio State University, Columbus, OH, USA

    Güzin Bayraksan

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  1. Tito Homem-de-Mello
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  1. University of Maryland, College Park, Maryland, USA

    Michael C Fu

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Homem-de-Mello, T., Bayraksan, G. (2015). Stochastic Constraints and Variance Reduction Techniques. In: Fu, M. (eds) Handbook of Simulation Optimization. International Series in Operations Research & Management Science, vol 216. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-1384-8_9

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