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Solutions of a constrained Hermitian matrix-valued function optimization problem with applications. (English) Zbl 1357.65079
The paper considers a group of problems on ranks/inertias, equalities/inequalities, minimization/maximization in the Löwner partial ordering of a Hermitian matrix-valued function subject to a linear matrix equation by using algebraic operations of the given matrices in the function and restriction. The author presents results in solving problems on Hermitian matrix-valued functions subject to linear matrix equation restrictions where the Lagrangian method is not available.

##### MSC:
 65K05 Numerical mathematical programming methods 15A09 Theory of matrix inversion and generalized inverses 15A24 Matrix equations and identities 47A62 Equations involving linear operators, with operator unknowns 90C22 Semidefinite programming
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##### References:
 [1] B. E. CAIN, E. M. DES ’A, The inertia of Hermitian matrices with a prescribed 2 $$\times$$ 2 block decom- position, Linear Multilinear Algebra 31 (1992), 119–130. [2] Y. CHABRILLAC, J. P. CROUZEIX, Definiteness and semidefiniteness of quadratic forms revisited, Linear Algebra Appl. 63 (1984), 283–292. · Zbl 0548.15027 [3] Y. CHEN, Nonnegative definite matrices and their applications to matrix quadratic programming prob- lems, Linear Multilinear Algebra 33 (1992), 189–201. [4] C. M. da Fonseca, The inertia of certain Hermitian block matrices, Linear Algebra Appl. 274 (1998), 193–210. · Zbl 0929.15019 [5] J. DANCIS, The possible inertias for a Hermitian matrix and its principal submatrices, Linear Algebra Appl. 85 (1987), 121–151. · Zbl 0614.15011 [6] J. DANCIS, Several consequences of an inertia theorem, Linear Algebra Appl. 136 (1990), 43–61. · Zbl 0712.15021 [7] E. V. HAYNSWORTH, Determination of the inertia of a partitioned Hermitian matrix, Linenr Algebra Appl. 1 (1968), 73–81. · Zbl 0155.06304 [8] E. V. HAYNSWORTH, A. M. OSTROWSKI, On the inertia of some classes of partitioned matrices, Linear Algebra Appl. 1 (1968), 299–316. · Zbl 0186.33704 [9] C. R. JOHNSON, L. RODMAN, Inertia possibilities for completions of partial Hermitian matrices, Linear Multilinear Algebra 16 (1984), 179–195. · Zbl 0548.15020 [10] Y. LIU, Y. TIAN, Max-min problems on the ranks and inertias of the matrix expressions ABXC $$\pm$$ (BXC) with applications, J. Optim. Theory Appl. 148 (2011), 593–622. · Zbl 1223.90077 [11] G. MARSAGLIA ANDG. P. H. STYAN, Equalities and inequalities for ranks of matrices, Linear Multilinear Algebra 2 (1974), 269–292. [12] R. PENROSE, A generalized inverse for matrices, Proc. Cambridge Phil. Soc. 51 (1955), 406–413. · Zbl 0065.24603 [13] S. PUNTANEN, G. P. H. STYAN, J. ISOTALO, Matrix Tricks for Linear Statistical Models, Our Per- sonal Top Twenty, Springer, 2011. · Zbl 1291.62014 [14] C. R. RAO, A lemma on optimization of matrix function and a review of the unified theory of linear estimation, In: Statistical Data Analysis and Inference, Y. Dodge (ed.), North Holland, pp. 397–417, 1989. · Zbl 0735.62066 [15] Y. TIAN, Equalities and inequalities for inertias of Hermitian matrices with applications, Linear Algebra Appl. 433 (2010), 263–296. · Zbl 1205.15033 [16] Y. TIAN, Maximization and minimization of the rank and inertia of the Hermitian matrix expression ABX (BX) with applications, Linear Algebra Appl. 434 (2011), 2109–2139. · Zbl 1211.15022 [17] Y. TIAN, Solving optimization problems on ranks and inertias of some constrained nonlinear matrix functions via an algebraic linearization method, Nonlinear Analysis 75 (2012), 717–734. · Zbl 1236.65070 [18] Y. TIAN, Formulas for calculating the extremum ranks and inertias of a four-term quadratic matrix- valued function and their applications, Linear Algebra Appl. 437 (2012), 835–859. · Zbl 1252.15026 [19] Y. TIAN, Solutions to 18 constrained optimization problems on the rank and inertia of the linear matrix function A+ BXB, Math. Comput. Modelling 55 (2012), 955–968. · Zbl 1255.15010 [20] Y. TIAN, Some optimization problems on ranks and inertias of matrix-valued functions subject to linear matrix equation restrictions, Banach J. Math. Anal. 8 (2014), 148–178. · Zbl 1278.15019 [21] Y. TIAN, A new derivation of BLUPs under random-effects model, Metrika 78 (2015), 905–918. · Zbl 1329.62264 [22] Y. TIAN, A matrix handling of predictions under a general linear random-effects model with new observations, Electron. J. Linear Algebra 29 (2015), 30–45. · Zbl 1329.62321 [23] Y. TIAN, A survey on rank and inertia optimization problems of the matrix-valued function A+BXB, Numer. Algebra Contr. Optim. 5 (2015), 289–326. · Zbl 1327.15007 [24] Y. TIAN, How to characterize properties of general Hermitian quadratic matrix-valued functions by rank and inertia, In: Advances in Linear Algebra Researches, I. Kyrchei, (ed.), Nova Publishers, New York, pp. 150–183, 2015.
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