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A primal–dual algorithm for risk minimization

Mathematical Programming volume 193, pages 337–363 (2022)Cite this article

Abstract

In this paper, we develop an algorithm to efficiently solve risk-averse optimization problems posed in reflexive Banach space. Such problems often arise in many practical applications as, e.g., optimization problems constrained by partial differential equations with uncertain inputs. Unfortunately, for many popular risk models including the coherent risk measures, the resulting risk-averse objective function is nonsmooth. This lack of differentiability complicates the numerical approximation of the objective function as well as the numerical solution of the optimization problem. To address these challenges, we propose a primal–dual algorithm for solving large-scale nonsmooth risk-averse optimization problems. This algorithm is motivated by the classical method of multipliers and by epigraphical regularization of risk measures. As a result, the algorithm solves a sequence of smooth optimization problems using derivative-based methods. We prove convergence of the algorithm even when the subproblems are solved inexactly and conclude with numerical examples demonstrating the efficiency of our method.

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Authors and Affiliations

  1. Optimization and Uncertainty Quantification, MS-1320, Sandia National Laboratories, P.O. Box 5800, Albuquerque, NM, 87185-1320, USA

    Drew P. Kouri

  2. FB12 Mathematik und Informatik, Philipps-Universität Marburg, Hans-Meerwein-Straße 6, Marburg, 35032, Germany

    Thomas M. Surowiec

Authors
  1. Drew P. Kouri
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  2. Thomas M. Surowiec
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Correspondence to Drew P. Kouri.

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DPK’s research was sponsored by DARPA EQUiPS Grant SNL 014150709, AFOSR Grant F4FGA09135G001 and Sandia National Laboratories LDRD “Risk-Adaptive Experimental Design for High-Consequence Systems”. Sandia National Laboratories is a multimission laboratory managed and operated by National Technology and Engineering Solutions of Sandia, LLC., a wholly owned subsidiary of Honeywell International, Inc., for the U.S. Department of Energy’s National Nuclear Security Administration under Contract DE-NA0003525. This paper describes objective technical results and analysis. Any subjective views or opinions that might be expressed in the paper do not necessarily represent the views of the U.S. Department of Energy or the United States Government. TMS’s research was sponsored by the DFG Grant No. SU 963/1-1 “Generalized Nash Equilibrium Problems with Partial Differential Operators: Theory, Algorithms, and Risk Aversion”.

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Kouri, D.P., Surowiec, T.M. A primal–dual algorithm for risk minimization. Math. Program. 193, 337–363 (2022). https://doi.org/10.1007/s10107-020-01608-9

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  • DOI: https://doi.org/10.1007/s10107-020-01608-9

Keywords

  • Risk-averse optimization
  • Coherent risk measures
  • Stochastic optimization
  • Method of multipliers

Mathematics Subject Classification

  • 49M29
  • 49M37
  • 65K10
  • 90C15
  • 93E20