Spherical optimization with complex variablesfor computing US-eigenpairs

• Published in: Computational optimization and applications Vol. 65; no. 3; pp. 799 - 820
• Main Authors: Ni, Guyan, Bai, Minru
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.12.2016; Springer Nature B.V
• Subjects: Algorithms; Approximation; Complex variables; Convex analysis; Convex and Discrete Geometry; Eigenvalues; Management Science; Mathematical analysis; Mathematical functions; Mathematical models; Mathematics; Mathematics and Statistics; Operations Research; Operations Research/Decision Theory; Optimization; Optimization algorithms; Statistics; Studies; Symmetry; Tensors; High-order power method; Optimization with complex variables; Quantum entanglement; US-eigenpair; Symmetric complex tensor; Rank-one approximation; Z-eigenpair; 15A69; 15A18; 81P40
• ISSN: 0926-6003; 1573-2894
• DOI: 10.1007/s10589-016-9848-7

The aim of this paper is to compute unitary symmetric eigenpairs (US-eigenpairs) of high-order symmetric complex tensors, which is closely related to the best complex rank-one approximation of a symmetric complex tensor and quantum entanglement. It is also an optimization problem of real-valued functions with complex variables. We study the spherical optimization problem with complex variables including the first-order and the second-order Taylor polynomials, optimization conditions and convex functions of real-valued functions with complex variables. We propose an algorithm and show that it is guaranteed to approximate a US-eigenpair of a symmetric complex tensor. Moreover, if the number of US-eigenpair is finite, then the algorithm is convergent to a US-eigenpair. Numerical examples are presented to demonstrate the effectiveness of the proposed method in finding US-eigenpairs.