Paper Info
Title
Complex Matrix Decomposition and Quadratic Programming
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 Zeitschrift3 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.
Year
DOI
Venue
2007
10.1287/moor.1070.0268
Math. Oper. Res.
Keywords
DocType
Volume
complex co-positivity cone,complex Hermitian matrix,quadratic optimization,Complex Matrix Decomposition,matrix rank-one decomposition,linear matrix inequality characterization,nonnegative quadratic function,J. F.,specific domain,linear matrix inequality,new matrix co-positive cone,Quadratic Programming,s-procedure.,S. Zhang,matrix rank-one decomposition result,matrix cone
Journal
32
Issue
ISSN
Citations 
3
0364-765X
88
PageRank 
References 
Authors
7.34
3
2
Name
Order
Citations
PageRank
Yongwei Huang181450.83
Shuzhong Zhang22808181.66