  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.

Optimization and Control;Numerical Analysis