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

Abstract

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 − X P X ∗ with respect to the variable matrix X by using a linearization method and some known formulas for extremum ranks and inertias of linear Hermitian matrix functions, where both P and Q are complex Hermitian matrices and X ∗ is the conjugate transpose of X . We then derive the global maximum and minimum ranks and inertias of the two quadratic Hermitian matrix functions ϕ 1 ( X ) = Q 1 − X P 1 X ∗ and ϕ 2 ( X ) = Q 2 − X ∗ P 2 X subject to a consistent matrix equation A X = 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 A X = B to satisfy the quadratic Hermitian matrix equalities X P 1 X ∗ = Q 1 and X ∗ P 2 X = Q 2 , respectively, and for the quadratic matrix inequalities X P 1 X ∗ > ( ⩾ , < , ⩽ ) Q 1 and X ∗ P 2 X > ( ⩾ , < , ⩽ ) Q 2 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 ϕ 1 ( X ) and ϕ 2 ( X ) subject to A X = B . Examples are also presented to illustrative applications of the equality-constrained quadratic optimizations in some matrix completion problems.