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Epi-Regularization of Risk Measures

Author

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  • Drew P. Kouri

    (Optimization and Uncertainty Quantification, Sandia National Laboratories, Albuquerque, New Mexico 87185;)

  • Thomas M. Surowiec

    (Fachbereich 12 Mathematik und Informatik, Philipps-Universität Marburg, 35037 Marburg, Germany)

Abstract

Uncertainty pervades virtually every branch of science and engineering, and in many disciplines, the underlying phenomena can be modeled by partial differential equations (PDEs) with uncertain or random inputs. This work is motivated by risk-averse stochastic programming problems constrained by PDEs. These problems are posed in infinite dimensions, which leads to a significant increase in the scale of the (discretized) problem. In order to handle the inherent nonsmoothness of, for example, coherent risk measures and to exploit existing solution techniques for smooth, PDE-constrained optimization problems, we propose a variational smoothing technique called epigraphical (epi-)regularization. We investigate the effects of epi-regularization on the axioms of coherency and prove differentiability of the smoothed risk measures. In addition, we demonstrate variational convergence of the epi-regularized risk measures and prove the consistency of minimizers and first-order stationary points for the approximate risk-averse optimization problem. We conclude with numerical experiments confirming our theoretical results.

Suggested Citation

  • Drew P. Kouri & Thomas M. Surowiec, 2020. "Epi-Regularization of Risk Measures," Mathematics of Operations Research, INFORMS, vol. 45(2), pages 774-795, May.
  • Handle: RePEc:inm:ormoor:v:45:y:2020:i:2:p:774-795
    DOI: 10.1287/moor.2019.1013
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    References listed on IDEAS

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    1. Andrzej Ruszczyński & Alexander Shapiro, 2006. "Optimization of Convex Risk Functions," Mathematics of Operations Research, INFORMS, vol. 31(3), pages 433-452, August.
    2. Aharon Ben‐Tal & Marc Teboulle, 2007. "An Old‐New Concept Of Convex Risk Measures: The Optimized Certainty Equivalent," Mathematical Finance, Wiley Blackwell, vol. 17(3), pages 449-476, July.
    3. Aharon Ben-Tal & Marc Teboulle, 1986. "Expected Utility, Penalty Functions, and Duality in Stochastic Nonlinear Programming," Management Science, INFORMS, vol. 32(11), pages 1445-1466, November.
    4. Philippe Artzner & Freddy Delbaen & Jean‐Marc Eber & David Heath, 1999. "Coherent Measures of Risk," Mathematical Finance, Wiley Blackwell, vol. 9(3), pages 203-228, July.
    5. Hans Föllmer & Alexander Schied, 2002. "Convex measures of risk and trading constraints," Finance and Stochastics, Springer, vol. 6(4), pages 429-447.
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    Cited by:

    1. Lee, Dongjin & Kramer, Boris, 2023. "Multifidelity conditional value-at-risk estimation by dimensionally decomposed generalized polynomial chaos-Kriging," Reliability Engineering and System Safety, Elsevier, vol. 235(C).

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