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

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.