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Implementable Algorithm for Stochastic Optimization Using Sample Average Approximations

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  • J. O. Royset

    (US Naval Postgraduate School)

  • E. Polak

    (University of California)

Abstract

We develop an implementable algorithm for stochastic optimization problems involving probability functions. Such problems arise in the design of structural and mechanical systems. The algorithm consists of a nonlinear optimization algorithm applied to sample average approximations and a precision-adjustment rule. The sample average approximations are constructed using Monte Carlo simulations or importance sampling techniques. We prove that the algorithm converges to a solution with probability one and illustrate its use by an example involving a reliability-based optimal design.

Suggested Citation

  • J. O. Royset & E. Polak, 2004. "Implementable Algorithm for Stochastic Optimization Using Sample Average Approximations," Journal of Optimization Theory and Applications, Springer, vol. 122(1), pages 157-184, July.
    • Handle: RePEc:spr:joptap:v:122:y:2004:i:1:d:10.1023_b:jota.0000041734.06199.71
    DOI: 10.1023/B:JOTA.0000041734.06199.71
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    References listed on IDEAS

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    1. Alan J. King & R. Tyrrell Rockafellar, 1993. "Asymptotic Theory for Solutions in Statistical Estimation and Stochastic Programming," Mathematics of Operations Research, INFORMS, vol. 18(1), pages 148-162, February.
    2. Kurt Marti, 1997. "Solving stochastic structural optimization problems by RSM-based stochastic approximation methods — gradient estimation in case of intermediate variables," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 46(3), pages 409-434, October.
    3. Julia L. Higle & Suvrajeet Sen, 1991. "Stochastic Decomposition: An Algorithm for Two-Stage Linear Programs with Recourse," Mathematics of Operations Research, INFORMS, vol. 16(3), pages 650-669, August.
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    Cited by:

    1. Rockafellar, R.T. & Royset, J.O., 2010. "On buffered failure probability in design and optimization of structures," Reliability Engineering and System Safety, Elsevier, vol. 95(5), pages 499-510.
    2. J. O. Royset & E. Polak, 2007. "Extensions of Stochastic Optimization Results to Problems with System Failure Probability Functions," Journal of Optimization Theory and Applications, Springer, vol. 133(1), pages 1-18, April.
    3. Wim Ackooij & Pedro Pérez-Aros, 2020. "Gradient Formulae for Nonlinear Probabilistic Constraints with Non-convex Quadratic Forms," Journal of Optimization Theory and Applications, Springer, vol. 185(1), pages 239-269, April.
    4. Johannes Royset, 2013. "On sample size control in sample average approximations for solving smooth stochastic programs," Computational Optimization and Applications, Springer, vol. 55(2), pages 265-309, June.

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