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Randomized Quasi-Monte Carlo Methods for Risk-Averse Stochastic Optimization

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Listed:
  • Olena Melnikov

    (Georgia Institute of Technology)

  • Johannes Milz

    (Georgia Institute of Technology)

Abstract

We establish epigraphical and uniform laws of large numbers for sample-based approximations of law invariant composite risk functionals. These sample-based approximation schemes include Monte Carlo and certain randomized quasi-Monte Carlo integration methods, such as scrambled net integration. Our results can be applied to the approximation of risk-averse stochastic programs and risk-averse stochastic variational inequalities. Our numerical simulations empirically demonstrate that randomized quasi-Monte Carlo approaches based on scrambled Sobol’ sequences can yield smaller bias and root mean square error than Monte Carlo methods for risk-averse optimization.

Suggested Citation

  • Olena Melnikov & Johannes Milz, 2025. "Randomized Quasi-Monte Carlo Methods for Risk-Averse Stochastic Optimization," Journal of Optimization Theory and Applications, Springer, vol. 206(1), pages 1-21, July.
    • Handle: RePEc:spr:joptap:v:206:y:2025:i:1:d:10.1007_s10957-025-02693-6
    DOI: 10.1007/s10957-025-02693-6
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    References listed on IDEAS

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    1. Lisa A. Korf & Roger J.-B. Wets, 2001. "Random LSC Functions: An Ergodic Theorem," Mathematics of Operations Research, INFORMS, vol. 26(2), pages 421-445, May.
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    3. H. Heitsch & H. Leövey & W. Römisch, 2016. "Are Quasi-Monte Carlo algorithms efficient for two-stage stochastic programs?," Computational Optimization and Applications, Springer, vol. 65(3), pages 567-603, December.
    4. Mihail Zervos, 1999. "On the Epiconvergence of Stochastic Optimization Problems," Mathematics of Operations Research, INFORMS, vol. 24(2), pages 495-508, May.
    5. Darinka Dentcheva & Spiridon Penev & Andrzej Ruszczyński, 2017. "Statistical estimation of composite risk functionals and risk optimization problems," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 69(4), pages 737-760, August.
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