Further results on sum-of-squares tensors

Abstract

Sum-of-squares (SOS) tensors plays an important role in tensor positive definiteness and polynomial optimization. So it is important to figure out what kind of tensors are SOS tensors. In this paper, we first show that several types of even order symmetric tensors are SOS tensors. The inclusive relation between several types of existing SOS tensors are developed under suitable conditions. By exploring the structure of LLK tensor, which is a special type of SOS tensor, we propose a numerical algorithm to check whether a given even order symmetric tensor is LLK tensor or not. Numerical experiments are also reported to show the efficiency of the proposed method.

Keywords:

• Symmetric tensor
• SOS tensor
• inclusive relation
• H-eigenvalue

Acknowledgments

We are very grateful to the editor and the two reviewers for their constructive comments and valuable suggestions on our manuscript.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The work was supported by the National Natural Science Foundation of China - NSFC (Grant No. 11601261, 11671228) and the Hong Kong Research Grants Council (Grant No. PolyU 15302114, 15300715, 15301716 and 15300717).

Notes on contributors

Haibin Chen

Haibin Chen received the BS degree from Qufu Normal University in 2004, the MS degree from Harbin Engineering University in 2006, and PhD degree from The Hong Kong Polytechnic University in 2016. He is currently a Associate Professor in Qufu Normal University. His research interests include Nonlinear optimization, Tensor optimization, and Tensor computing.

Liqun Qi

Liqun Qi received the BS degree from Tsinghua University in 1968, the MS degree from University of Wisconsin in 1981, and PhD degree from University of Wisconsin in 1984. He is now a Chair Professor of the Hong Kong Polytechnic University. His research interests include Optimization and Applications, Multi-linear Algebra, and Computational Mathematics and Application.

Yiju Wang

Yiju Wang received the BS and MS degrees from Qufu Normal University in 1990 and 1993 and received the PhD degree from Chinese Academy of Sciences in 2001. He is now a Professor of Qufu Normal University. His research interests include Nonlinear Optimization Algorithms, Sparse Optimization Theory and Tensor Optimization.

Guanglu Zhou

Guanglu Zhou received BS degree from Shandong Normal University, China in 1988, the MS degree from Qufu Normal University, China in 1992, and PhD from the University of New South Wales, Australia in 2000. He is Associate Professor of Curtin University. His research interests include Numerical Optimization, Computational Methods, Tensor Computation, Industrial Applications.