Spherical optimization with complex variablesfor computing US-eigenpairs

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• 作者：Guyan Ni ; Minru Bai
• 关键词：Symmetric complex tensor ; Rank ; one approximation ; US ; eigenpair ; Z ; eigenpair ; High ; order power method ; Quantum entanglement ; Optimization with complex variables
• 刊名：Computational Optimization and Applications
• 出版年：2016
• 出版时间：December 2016
• 年：2016
• 卷：65
• 期：3
• 页码：799-820
• 全文大小：604 KB
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Optimization
  Operations Research and Mathematical Programming
  Operation Research and Decision Theory
  Statistics
  Convex and Discrete Geometry
  
• 出版者：Springer Netherlands
• ISSN：1573-2894
• 卷排序：65

文摘

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.