Paper Info
Title
New results on Hermitian matrix rank-one decomposition
Abstract
In this paper, we present several new rank-one decomposition theorems for Hermitian positive semidefinite matrices, which generalize our previous results in Huang and Zhang (Math Oper Res 32(3):758–768, 2007), Ai and Zhang (SIAM J Optim 19(4):1735–1756, 2009). The new matrix rank-one decomposition theorems appear to have wide applications in theory as well as in practice. On the theoretical side, for example, we show how to further extend some of the classical results including a lemma due to Yuan (Math Program 47:53–63, 1990), the classical results on the convexity of the joint numerical ranges (Pang and Zhang in Unpublished Manuscript, 2004; Au-Yeung and Poon in Southeast Asian Bull Math 3:85–92, 1979), and the so-called Finsler’s lemma (Bohnenblust in Unpublished Manuscript; Au-Yeung and Poon in Southeast Asian Bull Math 3:85–92, 1979). On the practical side, we show that the new results can be applied to solve two typical problems in signal processing and communication: one for radar code optimization and the other for robust beamforming. The new matrix decomposition theorems are proven by construction in this paper, and we demonstrate that the constructive procedures can be implemented efficiently, stably, and accurately. The URL of our Matlab programs is given in this paper. We strongly believe that the new decomposition procedures, as a means to solve non-convex quadratic optimization with a few quadratic constraints, are useful for many other potential engineering applications.
Year
DOI
Venue
2011
10.1007/s10107-009-0304-7
Math. Program.
Keywords
Field
DocType
quadratic optimization.,new matrix rank-one decomposition,new decomposition procedure,new rank-one decomposition theorem,matrix rank-one decomposition,southeast asian bull math,new matrix decomposition theorem,positive semideflnite hermitian matrix,math program,new result,joint numer- ical range,classical result,math oper res,unpublished manuscript,matrix decomposition,quadratic optimization,numerical range,code optimization,signal processing,hermitian matrix
Discrete mathematics,Mathematical optimization,Convexity,Matrix (mathematics),Positive-definite matrix,Matrix decomposition,Complementarity theory,Quadratic programming,Hermitian matrix,Lemma (mathematics),Mathematics
Journal
Volume
Issue
ISSN
128
1-2
1436-4646
Citations 
PageRank 
References 
21
0.92
14
Authors
3
Name
Order
Citations
PageRank
Wen-Bao Ai11066.94
Yongwei Huang281450.83
Shuzhong Zhang32808181.66