- Solutions of a constrained Hermitian matrix-valued function optimization problem with applications Tian Yongge(yongge.tiangmail.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 ) Citation: Download: [PDF]Entry Submitted: 12/10/2015Entry Accepted: 12/10/2015Entry Last Modified: 12/10/2015Modify/Update this entry Visitors Authors More about us Links Subscribe, Unsubscribe Digest Archive Search, Browse the Repository Submit Update Policies Coordinator's Board Classification Scheme Credits Give us feedback Optimization Journals, Sites, Societies Optimization Online is supported by the Mathematical Optmization Society.