Complex matrix decomposition and quadratic programming

• 成果类型: Article
• 署名作者: Huang, Yongwei; Zhang, Shuzhong
• 署名单位: Chinese University of Hong Kong
• 刊物名称: MATHEMATICS OF OPERATIONS RESEARCH
• ISSN/ISSBN: 0364-765X
• DOI: 10.1287/moor.1070.0268
• 发表日期: 2007
• 页码: 758-768

摘要

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.