Epi-Regularization of Risk Measures

• 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: MATHEMATICS OF OPERATIONS RESEARCH
• ISSN/ISSBN: 0364-765X
• DOI: 10.1287/moor.2019.1013
• 发表日期: 2020
• 页码: 774-795
• 关键词: pde-constrained optimization; trust-region algorithm; stochastic collocation; Newton method; minimization

摘要

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 epl-regularization on the axioms of coherency and prove diffeientiability 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.