A primal-dual algorithm for risk minimization

• 成果类型: Article
• 署名作者: Kouri, Drew P.; Surowiec, Thomas M.
• 署名单位: United States Department of Energy (DOE); Sandia National Laboratories; Philipps University Marburg
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610
• DOI: 10.1007/s10107-020-01608-9
• 发表日期: 2022
• 页码: 337-363
• 关键词: pde-constrained optimization; trust-region algorithm; stochastic-approximation; collocation; descent

摘要

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.