On quantile cuts and their closure for chance constrained optimization problems

• 成果类型: Article
• 署名作者: Xie, Weijun; Ahmed, Shabbir
• 署名单位: Virginia Polytechnic Institute & State University; University System of Georgia; Georgia Institute of Technology
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610
• DOI: 10.1007/s10107-017-1190-z
• 发表日期: 2018
• 页码: 621-646
• 关键词: programs; sets

摘要

A chance constrained optimization problem over a finite distribution involves a set of scenario constraints from which a small subset can be violated. We consider the setting where all scenario constraints are mixed-integer convex. Existing works typically consider a mixed integer nonlinear programming (MINLP) formulation of this problem by introducing binary variables to indicate which constraint systems are to be satisfied or violated. A variety of cutting plane approaches for this MINLP formulation have been developed. In this paper we consider a family of cuts in the original space rather than those in the extended space of the MINLP reformulation. These cuts, known as quantile cuts, can be viewed as a projection of the well known family of mixing inequalities for the MINLP reformulation onto the original problem space. We show that the closure of the infinite family of all quantile cuts has a finite description. An important corollary of this result is that for linear chance constrained problems the quantile closure is polyhedral. We further show that a recursive application of quantile closure operations recovers the convex hull of the nonconvex chance constrained set in the limit, and in the pure integer setting the convergence is finite. We show that separation of quantile cuts is in general NP-hard, develop a heuristic separation method, and demonstrate its effectiveness through a computational study. We also study an approximation of the quantile closure and propose a generalization by grouping scenarios.