  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.
