Copositivity for a class of fourth order symmetric tensors given by scalar dark matter

In this paper, we mainly discuss the analytic expression of exact copositivity of 4th order symmetric tensor defined by the special physical model. We first show that for the general 4th order 2-dimensional symmetric tensor, it can be transformed into solving the quadratic polynomials, and then we give a necessary and sufficient condition to test the copositivity of 4th order 2-dimensional symmetric tensor. Based on this, we consider a special 4th order 3-dimensional symmetric tensor defined by the vacuum stability for $\mathbb{Z}_{3}$ scalar dark matter, and obtain the necessary and sufficient condition for its copositivity.Comment: 16 page

A continuation method for tensor complementarity problems

We introduce a Kojima-Megiddo-Mizuno type continuation method for solving tensor complementarity problems. We show that there exists a bounded continuation trajectory when the tensor is strictly semi-positive and any limit point tracing the trajectory gives a solution of the tensor complementarity problem. Moreover, when the tensor is strong strictly semi-positive, tracing the trajectory will converge to the unique solution. Some numerical results are given to illustrate the effectiveness of the method.Comment: 15 pages, 4 table