Analytical solutions to some optimization problems on ranks and inertias of matrix-valued functions subject to linear matrix inequalities

TL;DR

This paper derives explicit formulas for optimizing the rank and inertia of matrix functions under linear matrix inequality constraints, advancing understanding of matrix optimization problems.

Contribution

It provides explicit formulas for the maximum and minimum rank and inertia of certain Hermitian matrix expressions under linear matrix inequality constraints.

Findings

Formulas for maximal and minimal rank and inertia values

Characterization of constrained matrix-valued functions

Applications in matrix behavior analysis

Abstract

Matrix rank and inertia optimization problems are a class of discontinuous optimization problems, in which the decision variables are matrices running over certain feasible 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 linear matrix inequality $B_2XB_2^{*} \succcurlyeq A_2$ $(B_2XB_2^{*} \preccurlyeq A_2)$ in the L\"owner partial ordering, and give applications of these formulas in characterizing behaviors of some constrained matrix-valued functions.