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A survey on rank and inertia optimization problems of the matrix-valued function $$A+BXB^\ast$$. (English) Zbl 1327.15007
Summary: This paper is concerned with some rank and inertia optimization problems of the Hermitian matrix-valued functions $$A + BXB^{*}$$ subject to restrictions. We first establish several groups of explicit formula for calculating the maximum and minimum ranks and inertias of matrix sum $$A + X$$ subject to a Hermitian matrix $$X$$ that satisfies a fixed-rank and semi-definiteness restrictions by using some discrete and matrix decomposition methods. We then derive formulas for calculating the maximum and minimum ranks and inertias of the matrix-valued function $$A + BXB^*$$ subject to a Hermitian matrix $$X$$ that satisfies a fixed-rank and semi-definiteness restrictions, and show various properties $$A + BXB^{*}$$ from these ranks and inertias formulas. In particular, we give necessary and sufficient conditions for the equality $$A + BXB^* = 0$$ and the inequality $$A + BXB^* \succ 0\, (\succeq 0, \prec 0, \, \preceq 0)$$ to hold respectively for these specified Hermitian matrices $$X$$.

MSC:
 15A09 Theory of matrix inversion and generalized inverses 15A24 Matrix equations and identities 65K10 Numerical optimization and variational techniques 65K15 Numerical methods for variational inequalities and related problems
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