Models and algorithms for distributionally robust least squares problems

• 成果类型: Article
• 署名作者: Mehrotra, Sanjay; Zhang, He
• 署名单位: Northwestern University
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610
• DOI: 10.1007/s10107-013-0681-9
• 发表日期: 2014
• 页码: 123-141
• 关键词: optimization problems; stochastic programs; semidefinite

摘要

We present three different robust frameworks using probabilistic ambiguity descriptions of the data in least squares problems. These probability ambiguity descriptions are given by: (1) confidence region over the first two moments; (2) bounds on the probability measure with moments constraints; (3) the Kantorovich probability distance from a given measure. For the first case, we give an equivalent formulation and show that the optimization problem can be solved using a semidefinite optimization reformulation or polynomial time algorithms. For the second case, we derive the equivalent Lagrangian problem and show that it is a convex stochastic programming problem. We further analyze three special subcases: (i) finite support; (ii) measure bounds by a reference probability measure; (iii) measure bounds by two reference probability measures with known density functions. We show that case (i) has an equivalent semidefinite programming reformulation and the sample average approximations of case (ii) and (iii) have equivalent semidefinite programming reformulations. For ambiguity description (3), we show that the finite support case can be solved by using an equivalent second order cone programming reformulation.