Formulas for calculating the extremal ranks and inertias of a matrix-valued function subject to matrix equation restrictions

TL;DR

This paper derives explicit formulas for the extremal ranks and inertias of a matrix-valued function under certain matrix equation constraints, advancing matrix optimization theory.

Contribution

It provides new explicit formulas for extremal ranks and inertias of Hermitian matrix expressions with specific matrix equation restrictions.

Findings

Formulas for maximal and minimal rank and inertia values are established.

Conditions for the existence of common Hermitian solutions to matrix equations are derived.

Necessary and sufficient conditions for solution existence are provided.

Abstract

Matrix rank and inertia optimization problems are a class of discontinuous optimization problems in which the decision variables are matrices running over certain matrix sets, while the ranks and inertias of the variable matrices are taken as integer-valued objective functions. In this paper, we establish a group of explicit formulas for calculating the maximal and minimal values of the rank and inertia objective functions of the Hermitian matrix expression $A_1 - B_1XB_1^{*}$ subject to the common Hermitian solution of a pair of consistent matrix equations $B_2XB^{*}_2 = A_2$ and $B_3XB_3^{*} = A_3$, and Hermitian solution of the consistent matrix equation $B_4X= A_4$, respectively. Many consequences are obtained, in particular, necessary and sufficient conditions are established for the triple matrix equations $B_1XB^{*}_1 =A_1$, $B_2XB^{*}_2 = A_2$ and $B_3XB^{*}_3 = A_3$ to have a common Hermitian solution, as necessary and sufficient conditions for the two matrix equations $B_1XB^{*}_1 =A_1$ and $B_4X = A_4$ to have a common Hermitian solution.