Distributionally robust polynomial chance-constraints under mixture ambiguity sets

• 成果类型: Article
• 署名作者: Lasserre, Jean B.; Weisser, Tillmann
• 署名单位: Universite de Toulouse; Centre National de la Recherche Scientifique (CNRS); Universite de Toulouse; United States Department of Energy (DOE); Los Alamos National Laboratory; United States Department of Energy (DOE); Los Alamos National Laboratory
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610
• DOI: 10.1007/s10107-019-01434-8
• 发表日期: 2021
• 页码: 409-453
• 关键词: convex computation; optimization; uncertainty

摘要

Given X. Rn, e. (0, 1), a parametrized family of probability distributions (mu a)a.A on similar to. Rp, we consider the feasible set X* e. X associated with the distributionally robust chance-constraint X* e = {x. X : Prob mu[ f (x,.) > 0] > 1 - e,. mu. Ma}, whereMa is the set of all possibles mixtures of distributions mu a, a. A. For instance and typically, the family Ma is the set of all mixtures of Gaussian distributions on R with mean and standard deviation a = (a, s) in some compact set A. R2. We provide a sequence of inner approximationsXd e = {x. X : wd (x) < e}, d. N, where wd is a polynomial of degree d whose vector of coefficients is an optimal solution of a semidefinite program. The size of the latter increases with the degree d. We also obtain the strong and highly desirable asymptotic guarantee that.(X* e \Xd e). 0 as d increases, where. is the Lebesgue measure on X. Same results are also obtained for the more intricated case of distributionally robust joint chance-constraints. There is a price to pay for this strong asymptotic guarantee which is the scalability of such a numerical scheme, and so far this important drawback makes it limited to problems of modest dimension.