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Journal of Optimization Theory and Applications

, Volume 183, Issue 2, pp 365–385 | Cite as

Tensor Complementarity Problems—Part II: Solution Methods

  • Liqun QiEmail author
  • Zheng-Hai Huang
Invited Paper
  • 191 Downloads

Abstract

This work, with its three parts, reviews the state-of-the-art of studies for the tensor complementarity problem and some related models. In the first part of this paper, we have reviewed the theoretical developments of the tensor complementarity problem and related models. In this second part, we review the developments of solution methods for the tensor complementarity problem. It has been shown that the tensor complementarity problem is equivalent to some known optimization problems, or related problems such as systems of tensor equations, systems of nonlinear equations, and nonlinear programming problems, under suitable assumptions. By solving these reformulated problems with the help of structures of the involved tensors, several numerical methods have been proposed so that a solution of the tensor complementarity problem can be found. Moreover, based on a polynomial optimization model, a semidefinite relaxation method is presented so that all solutions of the tensor complementarity problem can be found under the assumption that the solution set of the problem is finite. Further applications of the tensor complementarity problem will be given and discussed in the third part of this paper.

Keywords

Tensor complementarity problem System of tensor equations System of non-smooth equations Mixed integer programming Semidefinite relaxation method 

Mathematics Subject Classification

90C33 90C30 65H10 

Notes

Acknowledgements

We are very grateful to professors Chen Ling, Yisheng Song, Shenglong Hu and Ziyan Luo for reading the first draft of this paper and putting forward valuable suggestions for revision. The first author’s work is partially supported by the Hong Kong Research Grant Council (Grant Nos. PolyU 15302114, 15300715, 15301716 and 15300717), and the second author’s work is partially supported by the National Natural Science Foundation of China (Grant Nos. 11431002 and 11871051).

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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Mathematics, School of ScienceHangzhou Dianzi UniversityHangzhouPeople’s Republic of China
  2. 2.Department of Applied MathematicsThe Hong Kong Polytechnic UniversityHung Hom, KowloonHong Kong
  3. 3.School of MathematicsTianjin UniversityTianjinPeople’s Republic of China

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