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

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 $A_1-B_{1}X{B}_1^{\ast }$ subject to the common Hermitian solution of a pair of consistent matrix equations $B_{2}X{B}_2^{\ast }=A_2$ and $B_{3}X{B}_3^{\ast }=A_3$, and Hermitian solution of the consistent matrix equation $B_{4}X=A_4$, respectively. Many consequences are obtained, in particular, necessary and sufficient conditions are established for the triple matrix equations $B_{1}X{B}_1^{\ast }=A_1$, $B_{2}X{B}_2^{\ast }=A_2$ and $B_{3}X{B}_3^{\ast }=A_3$ to have a common Hermitian solution, as necessary and sufficient conditions for the two matrix equations $B_{1}X{B}_1^{\ast }=A_1$ and $B_{4}X=A_4$ to have a common Hermitian solution.