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

Abstract

Let Ω={X∈Cn×p| AX=B, XH=K, and AA+B=B, KH+H=K, AK=BH}, 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∈Ω by applying the generalized inverses of matrices. Further, we present the solvability conditions for X∈Ω to satisfy the matrix equality (XC+D)M(XC+D)∗=G and matrix inequalities (XC+D)M(XC+D)∗> G(≥ G, < G, ≤ G)to hold, respectively. In addition, we provide closed-form solutions to two Löwner partial ordering optimization problems on f(X) subject to X∈Ω. Mathematics Subject Classification (2010). Primary 15A03; Secondary 15A09; 15A24; 15A63.