Tensor Complementarity Problems—Part II: Solution Methods

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. KeywordsTensor complementarity problem • System of tensor equations • System of non-smooth equations • Mixed integer programming • Semidefinite relaxation method Mathematics Subject Classification 90C33 • 90C30 • 65H10 B Liqun Qi

A Smoothing Newton Method with a Mixed Line Search for Monotone Weighted Complementarity Problems

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