Lifts of Convex Sets and Cone Factorizations

• 成果类型: Article
• 署名作者: Gouveia, Joao; Parrilo, Pablo A.; Thomas, Rekha R.
• 署名单位: Universidade de Coimbra; Massachusetts Institute of Technology (MIT); University of Washington; University of Washington Seattle
• 刊物名称: MATHEMATICS OF OPERATIONS RESEARCH
• ISSN/ISSBN: 0364-765X
• DOI: 10.1287/moor.1120.0575
• 发表日期: 2013
• 页码: 248-264
• 关键词: extended formulations; nonnegative rank; relaxations; PROGRAMS; optimization; complexity; matrices

摘要

In this paper, we address the basic geometric question of when a given convex set is the image under a linear map of an affine slice of a given closed convex cone. Such a representation or lift of the convex set is especially useful if the cone admits an efficient algorithm for linear optimization over its affine slices. We show that the existence of a lift of a convex set to a cone is equivalent to the existence of a factorization of an operator associated to the set and its polar via elements in the cone and its dual. This generalizes a theorem of Yannakakis that established a connection between polyhedral lifts of a polytope and nonnegative factorizations of its slack matrix. Symmetric lifts of convex sets can also be characterized similarly. When the cones live in a family, our results lead to the definition of the rank of a convex set with respect to this family. We present results about this rank in the context of cones of positive semidefinite matrices. Our methods provide new tools for understanding cone lifts of convex sets.