Complex Matrix Decomposition and Quadratic Programming
This paper studies the possibilities of the linear matrix inequality characterization of the matrix cones formed by nonnegative complex Hermitian quadratic functions over specific domains in the complex space. In its real-case analog, such studies were conducted in Sturm and Zhang [Sturm, J. F., S....
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| Veröffentlicht in: | Mathematics of operations research Jg. 32; H. 3; S. 758 - 768 |
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| Sprache: | Englisch |
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Linthicum
INFORMS
01.08.2007
Institute for Operations Research and the Management Sciences |
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| Abstract | This paper studies the possibilities of the linear matrix inequality characterization of the matrix cones formed by nonnegative complex Hermitian quadratic functions over specific domains in the complex space. In its real-case analog, such studies were conducted in Sturm and Zhang [Sturm, J. F., S. Zhang. 2003. On cones of nonnegative quadratic functions. Math. Oper. Res. 28 246–267]. In this paper it is shown that stronger results can be obtained for the complex Hermitian case. In particular, we show that the matrix rank-one decomposition result of Sturm and Zhang [Sturm, J. F., S. Zhang. 2003. On cones of nonnegative quadratic functions. Math. Oper. Res. 28 246–267] can be strengthened for the complex Hermitian matrices. As a consequence, it is possible to characterize several new matrix co-positive cones (over specific domains) by means of linear matrix inequality. As examples of the potential application of the new rank-one decomposition result, we present an upper bound on the lowest rank among all the optimal solutions for a standard complex semidefinite programming (SDP) problem, and offer alternative proofs for a result of Hausdorff [Hausdorff, F. 1919. Der Wertvorrat einer Bilinearform. Mathematische Zeitschrift 3 314–316] and a result of Brickman [Brickman, L. 1961. On the field of values of a matrix. Proc. Amer. Math. Soc. 12 61–66] on the joint numerical range. |
|---|---|
| AbstractList | This paper studies the possibilities of the linear matrix inequality characterization of the matrix cones formed by nonnegative complex Hermitian quadratic functions over specific domains in the complex space. In its real-case analog, such studies were conducted in Sturm and Zhang [Sturm, J. F., S. Zhang. 2003. On cones of nonnegative quadratic functions.
Math. Oper. Res.
28
246-267]. In this paper it is shown that stronger results can be obtained for the complex Hermitian case. In particular, we show that the matrix rank-one decomposition result of Sturm and Zhang [Sturm, J. F., S. Zhang. 2003. On cones of nonnegative quadratic functions.
Math. Oper. Res.
28
246-267] can be strengthened for the complex Hermitian matrices. As a consequence, it is possible to characterize several new matrix co-positive cones (over specific domains) by means of linear matrix inequality. As examples of the potential application of the new rank-one decomposition result, we present an upper bound on the lowest rank among all the optimal solutions for a standard complex semidefinite programming (SDP) problem, and offer alternative proofs for a result of Hausdorff [Hausdorff, F. 1919. Der Wertvorrat einer Bilinearform.
Mathematische Zeitschrift
3
314-316] and a result of Brickman [Brickman, L. 1961. On the field of values of a matrix.
Proc. Amer. Math. Soc.
12
61-66] on the joint numerical range. This paper studies the possibilities of the linear matrix inequality characterization of the matrix cones formed by nonnegative complex Hermitian quadratic functions over specific domains in the complex space. In its real-case analog, such studies were conducted in Sturm and Zhang [Sturm, J. F., S. Zhang. 2003. On cones of nonnegative quadratic functions. Math. Oper. Res. 28 246–267]. In this paper it is shown that stronger results can be obtained for the complex Hermitian case. In particular, we show that the matrix rank-one decomposition result of Sturm and Zhang [Sturm, J. F., S. Zhang. 2003. On cones of nonnegative quadratic functions. Math. Oper. Res. 28 246–267] can be strengthened for the complex Hermitian matrices. As a consequence, it is possible to characterize several new matrix co-positive cones (over specific domains) by means of linear matrix inequality. As examples of the potential application of the new rank-one decomposition result, we present an upper bound on the lowest rank among all the optimal solutions for a standard complex semidefinite programming (SDP) problem, and offer alternative proofs for a result of Hausdorff [Hausdorff, F. 1919. Der Wertvorrat einer Bilinearform. Mathematische Zeitschrift 3 314–316] and a result of Brickman [Brickman, L. 1961. On the field of values of a matrix. Proc. Amer. Math. Soc. 12 61–66] on the joint numerical range. This paper studies the possibilities of the linear matrix inequality characterization of the matrix cones formed by nonnegative complex Hermitian quadratic functions over specific domains in the complex space. In its real-case analog, such studies were conducted in Sturm and Zhang [Sturm, J. F., S. Zhang. 2003. On cones of nonnegative quadratic functions. Math. Oper. Res. 28 246-267]. In this paper it is shown that stronger results can be obtained for the complex Hermitian case. In particular, we show that the matrix rank-one decomposition result of Sturm and Zhang [Sturm, J. F., S. Zhang. 2003. On cones of nonnegative quadratic functions. Math. Oper. Res. 28 246-267] can be strengthened for the complex Hermitian matrices. As a consequence, it is possible to characterize several new matrix co-positive cones (over specific domains) by means of linear matrix inequality. As examples of the potential application of the new rank-one decomposition result, we present an upper bound on the lowest rank among all the optimal solutions for a standard complex semidefinite programming (SDP) problem, and offer alternative proofs for a result of Hausdorff [Hausdorff, F. 1919. Der Wertvorrat einer Bilinearform. Mathematische Zeitschrift 3 314-316] and a result of Brickman [Brickman, L. 1961. On the field of values of a matrix. Proc. Amer. Math. Soc. 12 61-66] on the joint numerical range. [PUBLICATION ABSTRACT] |
| Audience | Academic |
| Author | Huang, Yongwei Zhang, Shuzhong |
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| Cites_doi | 10.1287/moor.23.2.339 10.1007/BF02574037 10.1090/S0002-9939-1961-0122827-1 10.1137/S105262340139001X 10.1023/A:1021798932766 10.1287/moor.28.2.246.14485 10.1007/BF01292610 |
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| SubjectTerms | Analysis complex co-positivity cone Equation roots Inequalities (Mathematics) Linear equations Mathematical inequalities Mathematical optimization Mathematical sets Mathematical theorems Mathematical vectors Matrices Matrix Matrix decomposition matrix rank-one decomposition Operations research Optimal solutions Quadratic functions quadratic optimization Quadratic programming S-procedure Studies |
| Title | Complex Matrix Decomposition and Quadratic Programming |
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