On minimizing difference of a SOS-convex polynomial and a support function over a SOS-concave matrix polynomial constraint

• 成果类型: Article
• 署名作者: Lee, Jae Hyoung; Lee, Gue Myung
• 署名单位: Pukyong National University
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610
• DOI: 10.1007/s10107-017-1210-z
• 发表日期: 2018
• 页码: 177-198
• 关键词: optimization problems; set containments; semidefinite representation; dc optimization; Duality; PROGRAMS; INEQUALITY; squares; sums; relaxations

摘要

In this paper, we establish tractable sum of squares characterizations of the containment of a convex set, defined by a SOS-concave matrix inequality, in a non-convex set, defined by difference of a SOS-convex polynomial and a support function, with Slater's condition. Using our set containment characterization, we derive a zero duality gap result for a DC optimization problem with a SOS-convex polynomial and a support function, its sum of squares polynomial relaxation dual problem, the semidefinite representation of this dual problem, and the dual problem of the semidefinite programs. Also, we present the relations of their solutions. Finally, through a simple numerical example, we illustrate our results. Particularly, in this example we find the optimal solution of the original problem by calculating the optimal solution of its associated semidefinite problem.