Formulas for calculating the extremal ranks and inertias of a matrix-valued function subject to matrix equation restrictions
Provides theoretical tools for solving discontinuous optimization problems in matrix theory, but the contribution is incremental as it extends existing rank/inertia formulas to more complex constraints.
This paper derives explicit formulas for the extremal ranks and inertias of a Hermitian matrix expression under constraints from multiple matrix equations, and uses them to establish necessary and sufficient conditions for common Hermitian solutions to these equations.
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.