Polynomial Matrix Inequality and Semidefinite Representation
成果类型：
Article
署名作者：
Nie, Jiawang
署名单位：
University of California System; University of California San Diego
刊物名称：
MATHEMATICS OF OPERATIONS RESEARCH
ISSN/ISSBN：
0364-765X
DOI：
10.1287/moor.1110.0498
发表日期：
2011
页码：
398-415
关键词：
摘要：
Consider a convex set S = {x is an element of D: G(x) >= 0}, where G(x) is a symmetric matrix whose every entry is a polynomial or rational function, D subset of R-n is a domain on which G(x) is defined, and G(x) >= 0 means G(x) is positive semidefinite. The set S is called semidefinite representable if it equals the projection of a higher dimensional set that is defined by a linear matrix inequality (LMI). This paper studies sufficient conditions guaranteeing semidefinite representability of S. We prove that S is semidefinite representable in the following cases: (i) D = R-n, G(x) is a matrix polynomial and matrix sos-concave; (ii) D is compact convex, G(x) is a matrix polynomial and strictly matrix concave on D; (iii) G(x) is a matrix rational function and q-module matrix concave on D. Explicit constructions of semidefinite representations are given. Some examples are illustrated.