Polynomial Matrix Inequality and Semidefinite Representation

  • Authors:

  • Jiawang Nie

  • Affiliations:

  • Department of Mathematics, University of California at San Diego, La Jolla, California 92093

  • Venue:
  • Mathematics of Operations Research
  • Year:
  • 2011

Quantified Score

Hi-index 0.00

Visualization

Abstract

Consider a convex set $S=\{x \in \Cal{D}: G(x) \succeq 0\}$, where G(x) is a symmetric matrix whose every entry is a polynomial or rational function, D ⊆ Rn is a domain on which G(x) is defined, and $G(x) \succeq 0$ 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 = Rn, 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.