ALSO-X and ALSO-X1: Better Convex Approximations for Chance Constrained Programs

• 成果类型: Article
• 署名作者: Jiang, Nan; Xie, Weijun
• 署名单位: Virginia Polytechnic Institute & State University
• 刊物名称: OPERATIONS RESEARCH
• ISSN/ISSBN: 0030-364X
• DOI: 10.1287/opre.2021.2225
• 发表日期: 2022
• 页码: 3581-3600
• 关键词: chance constraint; CVaR; distributionally robust; bilevel optimization

摘要

In a chance constrained program (CCP), decision makers seek the best decision whose probability of violating the uncertainty constraints is within the prespecified risk level. As a CCP is often nonconvex and is difficult to solve to optimality, much effort has been devoted to developing convex inner approximations for a CCP, among which the conditional value-at-risk (CVaR) has been known to be the best for more than a decade. This paper studies and generalizes the ALSO-X, originally proposed by Ahmed, Luedtke, SOng, and Xie in 2017, for solving a CCP. We first show that the ALSO-X resembles a bilevel optimization, where the upper-level problemis to find the best objective function value and enforce the feasibility of a CCP for a given decision from the lower-level problem, and the lower-level problem is to minimize the expectation of constraint violations subject to the upper bound of the objective function value provided by the upper-level problem. This interpretation motivates us to prove that when uncertain constraints are convex in the decision variables, ALSO-X always outperforms the CVaR approximation. We further show (i) sufficient conditions under which ALSO-X can recover an optimal solution to a CCP; (ii) an equivalent bilinear programming formulation of a CCP, inspiring us to enhance ALSO-X with a convergent alternating minimization method (ALSO-X+); and (iii) an extension of ALSO-X and ALSO-X+ to distributionally robust chance constrained programs (DRCCPs) under the8-Wasserstein ambiguity set. Our numerical study demonstrates the effectiveness of the proposedmethods.