Exact Conic Programming Relaxations for a Class of Convex Polynomial Cone Programs

详细信息    查看全文

• 作者：Vaithilingam Jeyakumar ; Guoyin Li
• 关键词：Convex cone programs ; Convex polynomial optimization ; Conic programming ; Semidefinite programming
• 刊名：Journal of Optimization Theory and Applications
• 出版年：2017
• 出版时间：January 2017
• 年：2017
• 卷：172
• 期：1
• 页码：156-178
• 全文大小：
• 刊物主题：Calculus of Variations and Optimal Control; Optimization; Optimization; Theory of Computation; Applications of Mathematics; Engineering, general; Operation Research/Decision Theory;
• 出版者：Springer US
• ISSN：1573-2878
• 卷排序：172

文摘

In this paper, under a suitable regularity condition, we establish a broad class of conic convex polynomial optimization problems, called conic sum-of-squares convex polynomial programs, exhibiting exact conic programming relaxation, which can be solved by various numerical methods such as interior point methods. By considering a general convex cone program, we give unified results that apply to many classes of important cone programs, such as the second-order cone programs, semidefinite programs, and polyhedral cone programs. When the cones involved in the programs are polyhedral cones, we present a regularity-free exact semidefinite programming relaxation. We do this by establishing a sum-of-squares polynomial representation of positivity of a real sum-of-squares convex polynomial over a conic sum-of-squares convex system. In many cases, the sum-of-squares representation can be numerically checked via solving a conic programming problem. Consequently, we also show that a convex set, described by a conic sum-of-squares convex polynomial, is (lifted) conic linear representable in the sense that it can be expressed as (a projection of) the set of solutions to some conic linear systems.