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

• Published in: Nonlinear analysis Vol. 75; no. 2; pp. 717 - 734
• Main Author: Tian, Yongge
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier Ltd 2012; Elsevier
• Subjects: Algebra; Calculus of variations and optimal control; Exact sciences and technology; Finite differences and functional equations; Generalized inverse; Inertia; Linear and multilinear algebra, matrix theory; Linear matrix function; Linearization method; Löwner partial ordering; Mathematical analysis; Mathematics; Matrices; Matrix equation; Matrix inequality; Nonlinear algebraic and transcendental equations; Nonlinearity; Numerical analysis; Numerical analysis. Scientific computation; Optimization; Order disorder; Quadratic matrix function; Rank; Sciences and techniques of general use; 65K15; Löwner partial ordering; Rank; Matrix inequality; 15B57; 15A09; Generalized inverse; Linearization method; Optimization; 15A63; 15A24; 15B10; 65K10; Inertia; Quadratic matrix function; Linear matrix function; Matrix equation; Optimization method; Partial ordering; Lowner partial ordering; Mathematical model; Linearization; Nonlinear analysis; Matrix function; Algebraic method; Problem solving
• ISSN: 0362-546X; 1873-5215
• DOI: 10.1016/j.na.2011.09.003

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.