Solutions of a constrained Hermitian matrix-valued function optimization problem with applications
Yongge Tian · Operators and Matrices · 2016
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 X 0 of X 0 A = B such that the matrix inequality f (X) f (X 0 ) or f (X) f (X 0 ) 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 matrixvalued function and the corresponding semi-definiteness and optimization problems.