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Formulas for calculating the extremum ranks and inertias of a four-term quadratic matrix-valued function and their applications. (English) Zbl 1252.15026
Let \(A\) be an \(m\times m\) complex Hermitian matrix, \(B\) be an \(m\times n\) complex matrix, \(C\) be an \(n\times m\) complex Hermitian matrix, \(D\) be an \(n\times p\) complex matrix, \(X\) be a \(p\times m\) variable matrix and \((.)^{*}\) denotes the conjugate transpose of a complex matrix.
In this paper, the author presents a useful algebraic linearization method, which can convert the calculations of ranks and inertias of quadratic Hermitian matrix-valued functions (QHMF) into those of ranks and inertias of certain linear matrix-valued functions, then the author establishes a group of explicit formulas in closed form for calculating the global maximum and minimum ranks and inertias of this matrix-valued function with respect to the variable matrix \(X\). As applications of these rank and inertia formulas, the author characterizes a variety of solvability conditions for some quadratic matrix equations and inequalities generated from \(DXAX^{*}D^{*}+DXB+B^{*}X^{*}D^{*}+C\). In particular, the author gives analytical solutions to the two well-known classic optimization problems on the QHMF in the Löwner partial ordering. The results obtained and the techniques adopted for solving the matrix rank and inertia optimization problem enable us to make many new extensions of some classic results on quadratic forms, quadratic matrix equations and quadratic matrix inequalities.

MSC:
15A54 Matrices over function rings in one or more variables
15A24 Matrix equations and identities
15A63 Quadratic and bilinear forms, inner products
15B57 Hermitian, skew-Hermitian, and related matrices
90C20 Quadratic programming
90C22 Semidefinite programming
15B48 Positive matrices and their generalizations; cones of matrices
15A18 Eigenvalues, singular values, and eigenvectors
65K05 Numerical mathematical programming methods
15A45 Miscellaneous inequalities involving matrices
15A03 Vector spaces, linear dependence, rank, lineability
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References:
[1] Badawi, F.A., On a quadratic matrix inequality and the corresponding algebraic Riccati equation, Internat. J. contr., 6, 313-322, (1982) · Zbl 0486.93059
[2] Beck, A., Quadratic matrix programming, SIAM J. optim., 17, 1224-1238, (2006) · Zbl 1136.90025
[3] Beck, A., Convexity properties associated with nonconvex quadratic matrix-valued functions and applications to quadratic programming, J. optim. theory appl., 142, 1-29, (2009) · Zbl 1188.90190
[4] Belavkin, V.P.; Staszewski, P., Radon-Nikodym theorem for completely positive maps, Reports math. phys., 24, 49-55, (1986) · Zbl 0633.46060
[5] Boyd, S.; Vandenberghe, L., Convex optimization, (2004), Cambridge University Press · Zbl 1058.90049
[6] Calderón, A.P., A note on biquadratic forms, Linear algebra appl., 7, 175-177, (1973) · Zbl 0258.15016
[7] Candes, E.; Recht, B., Exact matrix completion via convex optimization, Found. comput. math., 9, 717-772, (2009) · Zbl 1219.90124
[8] Chabrillac, Y.; Crouzeix, J.P., Definiteness and semidefiniteness of quadratic forms revisited, Linear algebra appl., 63, 283-292, (1984) · Zbl 0548.15027
[9] Chen, Y., Nonnegative definite matrices and their applications to matrix quadratic programming problems, Linear and multilinear algebra, 33, 189-201, (1993) · Zbl 0764.15010
[10] Choi, M., Completely positive linear maps on complex matrices, Linear algebra appl., 10, 285-290, (1975) · Zbl 0327.15018
[11] Choi, M., Positive linear maps, Proc. sympos. pure math., 38, 583-590, (1982)
[12] Choi, M., Positive semidefinite biquadratic forms, Linear algebra appl., 12, 95-100, (1975) · Zbl 0336.15014
[13] Churilov, A.N., Solubility of matrix inequalities, Math. notes, 36, 862-866, (1984) · Zbl 0582.15006
[14] Cohen, N.; Dancis, J., Inertias of block band matrix completions, SIAM J. matrix anal. appl., 19, 583-612, (1998) · Zbl 0974.15009
[15] Crone, L., Second order adjoint matrix equations, Linear algebra appl., 39, 61-72, (1981) · Zbl 0464.15006
[16] de Klerk, E.; Pasechnik, D.V., Products of positive forms, linear matrix inequalities, and Hilbert 17th problem for ternary forms, Eur. J. oper. res., 157, 39-45, (2004) · Zbl 1106.90058
[17] de Pillis, J., Linear transformations which preserve Hermitian and positive semidefinite operators, Pacific J. math., 23, 129-137, (1967) · Zbl 0166.30003
[18] Dyn, N.; Ferguson, W.E., The numerical solution of equality-constrained quadratic programming problems, Math. comput., 41, 165-170, (1983) · Zbl 0527.49030
[19] M. Fazel, H. Hindi, S. Boyd, A rank minimization heuristic with application to minimum order system approximation, in: Proceedings of the 2001 American Control Conference, 2001, pp. 4734-4739.
[20] M. Fazel, H. Hindi, S. Boyd, Rank minimization and applications in system theory, in: Proceedings of the 2004 American Control Conference, 2004, pp. 3273-3278.
[21] Fujioka, H.; Hara, S., State covariance assignment problem with measurement noise: a unified approach based on a symmetric matrix equation, Linear algebra appl., 203-204, 579-605, (1994) · Zbl 0802.93044
[22] Galligani, E.; Zanni, L., Error analysis of an algorithm for equality-constrained quadratic programming problems, Computing, 58, 47-67, (1997) · Zbl 0865.65042
[23] Geelen, J.F., Maximum rank matrix completion, Linear algebra appl., 288, 211-217, (1999) · Zbl 0933.15026
[24] Gheondea, A., The three equivalent forms of completely positive maps on matrices, Ann. univ. bucharest (math. ser.), 59, 79-98, (2010) · Zbl 1224.46108
[25] Gill, P.E.; Murray, W.; Saunders, M.A.; Wright, M.H., Inertia-controlling methods for general quadratic programming, SIAM rev., 33, 1-36, (1991) · Zbl 0734.90062
[26] Gould, N.I.M., On practical conditions for the existence and uniqueness of solutions to the general equality quadratic programming problem, Math. program., 32, 90-99, (1985) · Zbl 0591.90068
[27] Gould, N.I.M.; Hribar, M.E.; Nocedal, J., On the solution of equality constrained quadratic programming problems arising in optimization, SIAM J. sci. comput., 23, 1376-1395, (2001) · Zbl 0999.65050
[28] N.J.A. Harvey, D.R. Karger, S. Yekhanin, The complexity of matrix completion, in: Proceedings of the Seventeenth Annual ACM-SIAM Symposium on Discrete Algorithm, Association for Computing Machinery, New York, 2006, pp. 1103-1111. · Zbl 1192.68322
[29] Hatcher, A., Algebraic topology, (2002), Cambridge University Press · Zbl 1044.55001
[30] Hoang, T.M.; Thierauf, T., The complexity of the inertia, (), 206-217 · Zbl 1027.68063
[31] T.M. Hoang, T. Thierauf, The complexity of the inertia and some closure properties of GapL, in: Proceedings of the Twentieth Annual IEEE Conference on Computational Complexity, 2005, pp. 28-37.
[32] James, I.J., The topology of Stiefel manifolds, (1976), Cambridge University Press · Zbl 0337.55017
[33] Jamshidi, M., An overview on the solutions of the algebraic matrix Riccati equation and related problems, Large scale syst., 1, 167-192, (1980) · Zbl 0453.93025
[34] Kawamoto, A.; Katayama, T., The semi-stabilizing solution of generalized algebraic Riccati equation for descriptor systems, Automatica, 38, 1651-1662, (2002) · Zbl 1011.93088
[35] Kawamoto, A.; Takaba, K.; Katayama, T., On the generalized algebraic Riccati equation for continuous-time descriptor systems, Linear algebra appl., 296, 1-4, (1999) · Zbl 0931.93059
[36] Khatskevich, V.A.; Karelin, I.I.; Zelenko, L., Operator pencils of the second order and linear fractional relations, Ukrain. math. bull., 3, 467-503, (2006) · Zbl 1148.47028
[37] Khatskevich, V.A.; Ostrovskii, M.I.; Shulman, V.S., Quadratic inequalities for Hilbert space operators, Integral equations operator theory, 59, 19-34, (2007) · Zbl 1133.47012
[38] Khatskevich, V.A.; Ostrovskii, M.I.; Shulman, V.S., Quadratic operator inequalities and linear-fractional relations, Funct. anal. appl., 41, 314-317, (2007) · Zbl 1155.47020
[39] Krafft, O., A matrix optimization problem, Linear algebra appl., 51, 137-142, (1983) · Zbl 0504.15013
[40] Laurent, M., Matrix completion problems, (), 221-229
[41] Li, C.-K.; Woerdeman, H.J., Special classes of positive and completely positive maps, Linear algebra appl., 255, 247-258, (1997) · Zbl 0874.15002
[42] Liu, Y.; Tian, Y., Max – min problems on the ranks and inertias of the matrix expressions \(A - \mathit{BXC} \pm(\mathit{BXC})^\ast\) with applications, J. optim. theory appl., 148, 593-622, (2011) · Zbl 1223.90077
[43] Ma, S.; Goldfarb, D.; Chen, L., Fixed point and Bregman iterative methods for matrix rank minimization, Math. program. ser. A, 128, 321-353, (2011) · Zbl 1221.65146
[44] Mahajan, M.; Sarma, J., On the complexity of matrix rank and rigidity, (), 269-280 · Zbl 1188.68158
[45] Masubuchi, I.; Kamitane, Y.; Ohara, A.; Suda, N., \(H_\infty\) control for descriptor systems: a matrix inequalities approach, Automatica, 33, 669-673, (1997) · Zbl 0881.93024
[46] Majthay, A., Optimality conditions for quadratic programming, Math. program., 1, 359-365, (1971) · Zbl 0246.90038
[47] Mesbahi, M., On the rank minimization problem and its control applications, Systems control lett., 33, 31-36, (1998) · Zbl 0902.93027
[48] M. Mesbahi, G.P. Papavassilopoulos, Solving a class of rank minimization problems via semi-definite programs, with applications to the fixed order output feedback synthesis, in: Proceedings of the American Control Conference, Albuquerque, New Mexico, 1997, pp. 77-80.
[49] Murty, K.G.; Kabadi, S.N., Some NP-complete problems in quadratic and nonlinear programming, Math. program., 39, 117-129, (1987) · Zbl 0637.90078
[50] Penrose, R., A generalized inverse for matrices, Proc. Cambridge philos. soc., 51, 406-413, (1955) · Zbl 0065.24603
[51] Razzaghi, M., Solution of the matrix Riccati equation in optimal control, Inform. sci., 16, 61-73, (1978) · Zbl 0443.93020
[52] Recht, B.; Fazel, M.; Parrilo, P.A., Guaranteed minimum rank solutions to linear matrix equations via nuclear norm minimization, SIAM rev., 52, 471-501, (2010) · Zbl 1198.90321
[53] Sayed, A.H.; Hassibi, B.; Kailath, T., Inertia properties of indefinite quadratic forms, IEEE sign. process. lett., 3, 57-59, (1996)
[54] Sayed, A.H.; Hassibi, B.; Kailath, T., Fundamental inertia conditions for the minimization of quadratic forms in indefinite metric spaces, Oper. theory adv. appl., 87, 309-447, (1996) · Zbl 0863.93091
[55] Stinespring, W.F., Positive functions on \(C^\ast\)-algebras, Proc. amer. math. soc., 6, 211-216, (1955) · Zbl 0064.36703
[56] Stømer, E., Extension of positive maps into \(B(H)\), J. funct. anal., 66, 235-254, (1986)
[57] Stømer, E., Separable states and positive maps, J. funct. anal., 254, 2303-2312, (2008) · Zbl 1143.46033
[58] Tian, Y., Equalities and inequalities for inertias of Hermitian matrices with applications, Linear algebra appl., 433, 263-296, (2010) · Zbl 1205.15033
[59] Tian, Y., Maximization and minimization of the rank and inertia of the Hermitian matrix expression \(A - \mathit{BX} -(\mathit{BX})^\ast\) with applications, Linear algebra appl., 434, 2109-2139, (2011) · Zbl 1211.15022
[60] Tian, Y., Extremal ranks of a quadratic matrix expression with applications, Linear and multilinear algebra, 59, 627-644, (2011) · Zbl 1220.15006
[61] Tian, Y., Solving optimization problems on ranks and inertias of some constrained nonlinear matrix functions via an algebraic linearization method, Nonlinear anal., 75, 717-734, (2012) · Zbl 1236.65070
[62] Tian, Y., Solutions to 18 constrained optimization problems on the rank and inertia of the linear matrix function \(A + \mathit{BXB}^\ast\), Math. comput. modelling, 55, 955-968, (2012) · Zbl 1255.15010
[63] VanAntwerp, J.G.; Braatz, R.D., A tutorial on linear and bilinear matrix inequalities, J. process. control, 10, 363-385, (2000)
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