An axiomatic duality framework for the theta body and related convex corners

• 成果类型: Article
• 署名作者: de Carli Silva, Marcel K.; Tuncel, Levent
• 署名单位: Universidade de Sao Paulo; University of Waterloo
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610
• DOI: 10.1007/s10107-016-1041-3
• 发表日期: 2017
• 页码: 283-323
• 关键词: perfect graph theorem; disjunctive index; chromatic number; Shannon capacity; optimization; delsarte; entropy; cones

摘要

Lovasz theta function and the related theta body of graphs have been in the center of the intersection of four research areas: combinatorial optimization, graph theory, information theory, and semidefinite optimization. In this paper, utilizing a modern convex optimization viewpoint, we provide a set of minimal conditions (axioms) under which certain key, desired properties are generalized, including the main equivalent characterizations of the theta function, the theta body of graphs, and the corresponding antiblocking duality relations. Our framework describes several semidefinite and polyhedral relaxations of the stable set polytope of a graph as generalized theta bodies. As a by-product of our approach, we introduce the notion of Schur Lifting of cones which is dual to PSD Lifting (more commonly used in SDP relaxations of combinatorial optimization problems) in our axiomatic generalization. We also generalize the notion of complements of graphs to diagonally scaling-invariant polyhedral cones. Finally, we provide a weighted generalization of the copositive formulation of the fractional chromatic number by Dukanovic and Rendl from 2010.