Optimization Under Probabilistic Envelope Constraints

• 成果类型: Article
• 署名作者: Xu, Huan; Caramanis, Constantine; Mannor, Shie
• 署名单位: National University of Singapore; University of Texas System; University of Texas Austin; Technion Israel Institute of Technology
• 刊物名称: OPERATIONS RESEARCH
• ISSN/ISSBN: 0030-364X
• DOI: 10.1287/opre.1120.1054
• 发表日期: 2012
• 页码: 682-699
• 关键词: robust solutions; Portfolio optimization; RISK; filters

摘要

Chance constraints are an important modeling tool in stochastic optimization, providing probabilistic guarantees that a solution succeeds in satisfying a given constraint. Although they control the probability of success, they provide no control whatsoever in the event of a failure. That is, they do not distinguish between a slight overshoot or undershoot of the bounds and more catastrophic violation. In short, they do not capture the magnitude of violation of the bounds. This paper addresses precisely this topic, focusing on linear constraints and ellipsoidal (Gaussian-like) uncertainties. We show that the problem of requiring different probabilistic guarantees at each level of constraint violation can be reformulated as a semi-infinite optimization problem. We provide conditions that guarantee polynomial-time solvability of the resulting semi-infinite formulation. We show further that this resulting problem is what has been called a comprehensive robust optimization problem in the literature. As a byproduct, we provide tight probabilistic bounds for comprehensive robust optimization. Thus, analogously to the connection between chance constraints and robust optimization, we provide a broader connection between probabilistic envelope constraints and comprehensive robust optimization.