The Optimization Problems of a Quadratic Hermitian Matrix-valued Function with the Constraint of Matrix Equations

Abstract

Let ${\rm \Omega}=\left\{X \in {\mathbb {C}} ^{n \times p} \left|\ AX=B,XH=K,~and~AA^\dagger B=B,KH^\dagger H=K,AK=BH\right.\right\},$ and let $f(X)=(XC+D)M(XC+D)^*-G$ be a given quadratic Hermitian matrix-valued function. In this paper, we first establish a series of closed-form formulas for calculating the extremal ranks and inertias of $f(X)$ subject to $X\in {\rm \Omega}$ by applying the generalized inverses of matrices. Further, we present the solvability conditions for $X\in {\rm \Omega}$ to satisfy the matrix equality $(XC+D)M(XC+D)^*=G$ and matrix inequalities $(XC+D)M(XC+D)^*>G(\geq G,<G,\leq G)$ to hold, respectively. In addition, we provide closed-form solutions to two L\"{o}wner partial ordering optimization problems on $f(X)$ subject to $X\in {\rm \Omega}$.