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

摘要

Matrix rank and inertia optimization problems are a class of discontinuousoptimization problems in which the decision variables are matrices running overcertain matrix sets, while the ranks and inertias of the variable matrices aretaken as integer-valued objective functions. In this paper, we establish agroup of explicit formulas for calculating the maximal and minimal values ofthe rank and inertia objective functions of the Hermitian matrix expression$A_1 - B_1XB_1^{*}$ subject to the common Hermitian solution of a pair ofconsistent matrix equations $B_2XB^{*}_2 = A_2$ and $B_3XB_3^{*} = A_3$, andHermitian solution of the consistent matrix equation $B_4X= A_4$, respectively.Many consequences are obtained, in particular, necessary and sufficientconditions 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 Hermitiansolution, 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.