S-lemma with equality and its applications

• 成果类型: Article
• 署名作者: Xia, Yong; Wang, Shu; Sheu, Ruey-Lin
• 署名单位: Beihang University; National Cheng Kung University
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610
• DOI: 10.1007/s10107-015-0907-0
• 发表日期: 2016
• 页码: 513-547
• 关键词: nonconvex quadratic optimization; Strong Duality; convexity; constraints

摘要

Let and be two quadratic functions having symmetric matrices and . The S-lemma with equality asks when the unsolvability of the system implies the existence of a real number such that . The problem is much harder than the inequality version which asserts that, under Slater condition, is unsolvable if and only if for some . In this paper, we show that the S-lemma with equality does not hold only when the matrix has exactly one negative eigenvalue and is a non-constant linear function (). As an application, we can globally solve as well as the two-sided generalized trust region subproblem without any condition. Moreover, the convexity of the joint numerical range where is a (possibly non-convex) quadratic function and are affine functions can be characterized using the newly developed S-lemma with equality.