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
Hauptverfasser: Huang, Yongwei, Zhang, Shuzhong
Format: Journal Article
Sprache:Englisch
Veröffentlicht: Linthicum INFORMS 01.08.2007
Institute for Operations Research and the Management Sciences
Schlagworte:
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
United States
ISSN:0364-765X, 1526-5471
Online-Zugang:Volltext
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Zusammenfassung: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.
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ISSN:0364-765X
1526-5471
DOI:10.1287/moor.1070.0268