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Solutions of a constrained Hermitian matrix-valued function optimization problem with applications

Tian Yongge(yongge.tian***at***gmail.com)

Abstract: Let $f(X) =\left( XC + D\right)M\left(XC + D \right)^{*} - G$ be a given nonlinear Hermitian matrix-valued function with $M = M^*$ and $G = G^*$, and assume that the variable matrix $X$ satisfies the consistent linear matrix equation $XA = B$. This paper shows how to characterize the semi-definiteness of $f(X)$ subject to all solutions of $XA = B$. As applications, a standard method is obtained for finding analytical solutions $X_0$ of $X_0A = B$ such that the matrix inequality $f(X) \succcurlyeq f(X_0)$ or $f(X) \preccurlyeq f(X_0)$ holds for all solutions of $X\!A = B$. The whole work provides direct access, as a standard example, to a very simple algebraic treatment of the constrained Hermitian matrix-valued function and the corresponding semi-definiteness and optimization problems.

Keywords: Hermitian matrix-valued function; rank, inertia, semi-definiteness; L\"owner partial ordering; optimization problem

Category 1: Nonlinear Optimization (Constrained Nonlinear Optimization )


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Entry Submitted: 12/10/2015
Entry Accepted: 12/10/2015
Entry Last Modified: 12/10/2015

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