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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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##### References:
  W. Ai, On the low rank solutions for linear matrix inequalities,, Math. Oper. Res., 33, 965, (2008) · Zbl 1218.90152  E. M. de Sá, On the inertia of sums of Hermitian matrices,, Linear Algebra Appl., 37, 143, (1981) · Zbl 0457.15012  D. A. Gregory, Inertia and biclique decompositions of joins of graphs,, J. Combin. Theory Ser. B, 88, 135, (2003) · Zbl 1025.05042  M. Journée, Low-rank optimization on the cone of positive semidefinite matrices,, SIAM J. Optim., 20, 2327, (2010) · Zbl 1215.65108  C.-K. Li, Sum of Hermitian matrices with given eigenvalues: inertia, rank, and multiple eigenvalues,, Canad. J. Math., 62, 109, (2010) · Zbl 1188.15039  Y. Liu, More on extremal ranks of the matrix expressions A-BX± X*B* with statistical applications,, Numer. Linear Algebra Appl., 15, 307, (2008) · Zbl 1212.15029  Y. Liu, Extremal ranks of submatrices in an Hermitian solution to the matrix equation AXA*= B with applications,, J. Appl. Math. Comput., 32, 289, (2010) · Zbl 1194.15014  Y. Liu, A simultaneous decomposition of a matrix triplet with applications,, Numer. Linear Algebra Appl., 18, 69, (2011) · Zbl 1249.15020  Y. Liu, Max-min problems on the ranks and inertias of the matrix expressions A-BXC ± (BXC)* with applications,, J. Optim. Theory Appl., 148, 593, (2011) · Zbl 1223.90077  Y. Liu, Hermitian-type of singular value decomposition for a pair of matrices and its applications,, Numer. Linear Algebra Appl., 20, 60, (2013) · Zbl 1289.65108  Y. Liu, Ranks of Hermitian and skew-Hermitian solutions to the matrix equation AXA=B*,, Linear Algebra Appl., 431, 2359, (2009) · Zbl 1180.15018  C. Lu, Revisit to the problem of generalized low rank approximation of matrices,, In: ICIC 2006 (D.-S. Huang, 345, 450, (2006) · Zbl 1202.65057  J. H. Manton, The geometry of weighted low-rank approximations,, IEEE Trans. Sign. Process., 51, 500, (2003) · Zbl 1369.94221  G. Marsaglia, Equalities and inequalities fo ranks of matrices,, Linear Multilinear Algebra, 2, 269, (1974)  D. V. Ouellette, Schur complements and statistics,, Linear Algebra Appl., 36, 187, (1981) · Zbl 0455.15012  R. E. Skelton, A Unified Algebraic Approach to Linear Control Design,, Taylor & Francis, (1997)  Y. Tian, Solvability of two linear matrix equations,, Linear Multilinear Algebra, 48, 123, (2000) · Zbl 0970.15005  Y. Tian, Equalities and inequalities for inertias of Hermitian matrices with applications,, Linear Algebra Appl., 433, 263, (2010) · Zbl 1205.15033  Y. Tian, Rank and inertia of submatrices of the Moore-Penrose inverse of a Hermitian matrix,, Electron. J. Linear Algebra, 20, 226, (2010) · Zbl 1207.15007  Y. Tian, Completing block Hermitian matrices with maximal and minimal ranks and inertias,, Electron. J. Linear Algebra, 21, 124, (2010) · Zbl 1207.15029  Y. Tian, Maximization and minimization of the rank and inertia of the Hermitian matrix expression A - BX - (BX)* with applications,, Linear Algebra Appl., 434, 2109, (2011) · Zbl 1211.15022  Y. Tian, Solutions to 18 constrained optimization problems on the rank and inertia of the linear matrix function A + BXB*,, Math. Comput. Modelling, 55, 955, (2012) · Zbl 1255.15010  Y. Tian, On additive decompositions of the Hermitian solutions of the matrix equation AXA*= B,, Mediterr. J. Math., 9, 47, (2012) · Zbl 1241.15012  Y. Tian, On an equality and four inequalities for generalized inverses of Hermitian matrices,, Electron. J. Linear Algebra, 23, 11, (2012) · Zbl 1253.15025  Y. Tian, Equalities and inequalities for Hermitian solutions and Hermitian definite solutions of the two matrix equations AX = B and AXA* = B,, Aequat. Math., 86, 107, (2013) · Zbl 1281.15016  Y. Tian, Some optimization problems on ranks and inertias of matrix-valued functions subject to linear matrix equation restrictions,, Banach J. Math. Anal., 8, 148, (2014) · Zbl 1278.15019  Y. Tian, Extremal ranks of some symmetric matrix expressions with applications,, SIAM J. Matrix Anal. Appl., 28, 890, (2006) · Zbl 1123.15001  J. Ye, Generalized low rank approximations of matrices,, Machine Learning, 61, 167, (2005) · Zbl 1087.65043  H. Zha, A note on the existence of the hyperbolic singular value decomposition,, Linear Algebra Appl., 240, 199, (1996) · Zbl 0923.15007
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