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Some optimization problems on ranks and inertias of matrix-valued functions subject to linear matrix equation restrictions. (English) Zbl 1278.15019
Summary: 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-valued function \(A_1 - B_1XB_1^{*}\) 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 the 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 well 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.

MSC:
15A24 Matrix equations and identities
15B57 Hermitian, skew-Hermitian, and related matrices
15A03 Vector spaces, linear dependence, rank, lineability
90C11 Mixed integer programming
90C22 Semidefinite programming
15A18 Eigenvalues, singular values, and eigenvectors
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