Solving optimization problems on ranks and inertias of some constrained nonlinear matrix functions via an algebraic linearization method

We establish in this paper a group of closed-form formulas for calculating the global maximum and minimum ranks and inertias of the quadratic Hermitian matrix function ϕ(X)=Q−XPX∗ϕ(X)=Q−XPX∗ with respect to the variable matrix XX by using a linearization method and some known formulas for extremum ranks and inertias of linear Hermitian matrix functions, where both PP and QQ are complex Hermitian matrices and X∗X∗ is the conjugate transpose of XX. We then derive the global maximum and minimum ranks and inertias of the two quadratic Hermitian matrix functions ϕ1(X)=Q1−XP1X∗ϕ1(X)=Q1−XP1X∗ and ϕ2(X)=Q2−X∗P2Xϕ2(X)=Q2−X∗P2X subject to a consistent matrix equation AX=BAX=B, respectively, by using some pure algebraic operations of matrices and their generalized inverses. As consequences, we establish necessary and sufficient conditions for the solutions of the matrix equation AX=BAX=B to satisfy the quadratic Hermitian matrix equalities XP1X∗=Q1XP1X∗=Q1 and X∗P2X=Q2X∗P2X=Q2, respectively, and for the quadratic matrix inequalities XP1X∗>(⩾,<,⩽)Q1 and X∗P2X>(⩾,<,⩽)Q2 in the Löwner partial ordering to hold, respectively. In addition, we give complete solutions to four Löwner partial ordering optimization problems on the matrix functions ϕ1(X)ϕ1(X) and ϕ2(X)ϕ2(X) subject to AX=BAX=B. Examples are also presented to illustrative applications of the equality-constrained quadratic optimizations in some matrix completion problems.