Epi-Regularization of Risk Measures

Drew P. Kouri () and Thomas M. Surowiec ()
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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

Mathematics of Operations Research, 2020, vol. 45, issue 2, 774-795

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.

Keywords: risk averse; coherent risk measures; uncertainty quantification; stochastic optimization; infimal convolution; PDE-constrained optimization (search for similar items in EconPapers)
Date: 2020
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Citations: View citations in EconPapers (1)

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Persistent link: https://EconPapers.repec.org/RePEc:inm:ormoor:v:45:y:2020:i:2:p:774-795

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