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A Stochastic Approximation Method for Simulation-Based Quantile Optimization

Author

Listed:
  • Jiaqiao Hu

    (Department of Applied Mathematics and Statistics, State University of New York at Stony Brook, Stony Brook, New York 11794)

  • Yijie Peng

    (Department of Management Science and Information Systems, Guanghua School of Management Peking University, Beijing 100871, China)

  • Gongbo Zhang

    (Department of Management Science and Information Systems, Guanghua School of Management Peking University, Beijing 100871, China)

  • Qi Zhang

    (Department of Applied Mathematics and Statistics, State University of New York at Stony Brook, Stony Brook, New York 11794)

Abstract

We present a gradient-based algorithm for solving a class of simulation optimization problems in which the objective function is the quantile of a simulation output random variable. In contrast with existing quantile (quantile derivative) estimation techniques, which aim to eliminate the estimator bias by gradually increasing the simulation sample size, our algorithm incorporates a novel recursive procedure that only requires a single simulation sample at each step to simultaneously obtain quantile and quantile derivative estimators that are asymptotically unbiased. We show that these estimators, when coupled with the standard gradient descent method, lead to a multitime-scale stochastic approximation type of algorithm that converges to an optimal quantile value with probability one. In our numerical experiments, the proposed algorithm is applied to optimal investment portfolio problems, resulting in new solutions that complement those obtained under the classical Markowitz mean-variance framework.

Suggested Citation

  • Jiaqiao Hu & Yijie Peng & Gongbo Zhang & Qi Zhang, 2022. "A Stochastic Approximation Method for Simulation-Based Quantile Optimization," INFORMS Journal on Computing, INFORMS, vol. 34(6), pages 2889-2907, November.
    • Handle: RePEc:inm:orijoc:v:34:y:2022:i:6:p:2889-2907
    DOI: 10.1287/ijoc.2022.1214
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    References listed on IDEAS

    as
    1. Yijie Peng & Michael C. Fu & Bernd Heidergott & Henry Lam, 2020. "Maximum Likelihood Estimation by Monte Carlo Simulation: Toward Data-Driven Stochastic Modeling," Operations Research, INFORMS, vol. 68(6), pages 1896-1912, November.
    2. Andrey Kibzun & Evgeniy Matveev, 2012. "Optimization of the quantile criterion for the convex loss function by a stochastic quasigradient algorithm," Annals of Operations Research, Springer, vol. 200(1), pages 183-198, November.
    3. Qi Zhang & Jiaqiao Hu, 2019. "Simulation Optimization Using Multi-Time-Scale Adaptive Random Search," Asia-Pacific Journal of Operational Research (APJOR), World Scientific Publishing Co. Pte. Ltd., vol. 36(06), pages 1-34, December.
    4. Guangwu Liu & Liu Jeff Hong, 2009. "Kernel estimation of quantile sensitivities," Naval Research Logistics (NRL), John Wiley & Sons, vol. 56(6), pages 511-525, September.
    5. L. Jeff Hong, 2009. "Estimating Quantile Sensitivities," Operations Research, INFORMS, vol. 57(1), pages 118-130, February.
    6. Bernd Heidergott & Warren Volk-Makarewicz, 2016. "A Measure-Valued Differentiation Approach to Sensitivities of Quantiles," Mathematics of Operations Research, INFORMS, vol. 41(1), pages 293-317, February.
    7. Michael C. Fu & L. Jeff Hong & Jian-Qiang Hu, 2009. "Conditional Monte Carlo Estimation of Quantile Sensitivities," Management Science, INFORMS, vol. 55(12), pages 2019-2027, December.
    8. Guangxin Jiang & Michael C. Fu, 2015. "Technical Note—On Estimating Quantile Sensitivities via Infinitesimal Perturbation Analysis," Operations Research, INFORMS, vol. 63(2), pages 435-441, April.
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