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

Abstract. Let f(X) = (XC + D )M(XC + D) − 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 X0 of X0A = B such that the matrix inequality f(X) < f(X0) or f(X) 4 f(X0) holds for all solutions of XA = 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.