## 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.

Original language | English |
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Pages (from-to) | 365-385 |

Number of pages | 21 |

Journal | Journal of Optimization Theory and Applications |

Volume | 183 |

Issue number | 2 |

DOIs | |

Publication status | Published - 1 Nov 2019 |

## Keywords

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

## ASJC Scopus subject areas

- Control and Optimization
- Management Science and Operations Research
- Applied Mathematics