A primal–dual algorithm for risk minimization

Kouri, Drew P.; Surowiec, Thomas M.

doi:10.1007/s10107-020-01608-9

A primal–dual algorithm for risk minimization

Journal Article · 04 February 2021 · Mathematical Programming

DOI:https://doi.org/10.1007/s10107-020-01608-9· OSTI ID:1765742

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Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)
Philipps-Univ. Marburg (Germany)

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. Here, 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.

View Accepted Manuscript (DOE)

Research Organization:: Sandia National Laboratories (SNL-NM), Albuquerque, NM (United States)

Sponsoring Organization:: USDOE National Nuclear Security Administration (NNSA); USDOE Laboratory Directed Research and Development (LDRD) Program; Defense Advanced Research Projects Agency (DARPA); US Air Force Office of Scientific Research (AFOSR); German Research Foundation (DFG)

Grant/Contract Number:: AC04-94AL85000; NA0003525

OSTI ID:: 1765742

Report Number(s):: SAND--2021-0818J; 693611

Journal Information:: Mathematical Programming, Journal Name: Mathematical Programming Journal Issue: 1 Vol. 193; ISSN 0025-5610

Publisher:: SpringerCopyright Statement

Country of Publication:: United States

Language:: English

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97 MATHEMATICS AND COMPUTING
risk-averse optimization
coherent risk measures
stochastic optimization
method of multipliers

A primal–dual algorithm for risk minimization

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