A Monotone Semismooth Newton Method for a Kind of Tensor Complementarity Problem

Tensor complementarity problem (TCP) is a special kind of nonlinear complementarity problem (NCP). In this paper, we introduce a new class of structure tensor and give some examples. By transforming the TCP to the system of nonsmooth equations, we develop a semismooth Newton method for the tensor complementarity problem. We prove the monotone convergence theorem for the proposed method under proper conditions.


Introduction
A Tensor is a multi-arrray. We denote by ( ) , m n T the space of all m-order and n-dimension tensors.

( )
, m n ∈ A T is in the form of In this paper, we discuss the following tensor complementarity problem (simplified as TCP ( ) , , A p q ): finding a point n x R ∈ such that ( ) ( ) ( ) is an n-dimensional vector, whose ith component is given by [1] ( ) Tensor complementarity problem has many applications, such as nonlinear compressed sensing, communications, DNA microarrays, n-person noncooperative game and so on, see for example [2] [3]. TCP has received much attention and has taken good progress in recent years [4]- [13], such as the structure of the solution set, the global uniqueness solvability and the error bound and so on.
Bai, Huang and Wang [4] proved that the P-tensor complementarity problem has a nonempty compact solution set. What's more, they showed the global uniqueness solvability property for the TCP with a strong P-tensor. Che, Qi and Wei [5] showed that when the tensor is a positive definite or strictly copositive tensor, the TCP has a nonempty compact solution set. On the other hand, however, the study in the related numerical methods is very few. Luo, Qi and Xiu [3] proposed an iterative method to find the sparsest solutions to the Z-tensor complementarity problem with a non-positive constant term. Xu, Li and Xie [14] concerned with the tensor complementarity problem with a positive semi-definite Z-tensor. Under the assumption that the problem has a solution at which the strict complementarity holds, they showed that the problem is equivalent to a system of lower dimensional tensor equations. In this paper, we present a se- Similarly, we denote I q as the r-dimensional subvector of q and its elements are i q , i I ∈ .

Semismooth Newton Method and Its Convergence
We first introduce a new class of structure tensor, called M-like tensor.
Remark 2.1. As an example, it is easy to verify that the even-order diagonal tensors with all positive diagonal entries are M-like tensors. We give a non-diagonal M-like tensor. It is easy to verify A is an M-like tensor over 2 R + . In the following, without specification, we always suppose A is an M-like tensor.
Let 2 : R R → φ be the well-known Fischer-Burmeister function defined by ( ) 2 2 , It is easy to see that TCP (1.1) can be transformed to the system of nonsmooth equations: can be expressed as follows.
It is easy to see that 0, 0  . Now, we present semismooth Newton method for TCP (1.1).
Step 1. If Step 2. Choose Step 3. Calculate and i e is the ith row of the n n × identity matrix. It is easy to verify V KB = , where where ii a is the ith diagonal element of ( ) .
For any x D ∈ , we have x p ≥ and Associated to x, we define two disjoint index sets as follows: x y ≤ and y D ∈ . Since Now, we prove Associated to x, we define four disjoint index sets as follows:

Conclusion
We have introduced a new kind of structure tensor and discussed the numerical algorithm for TCP. By transforming the TCP to the system of nonsmooth equations, we have presented a semismooth Newton method for TCP. At each iteration, only linear equations need to be solved. The sequence generated by the algorithm is monotonically convergent to the solution of the TCP under proper conditions. This method can be regarded as a kind of Newton-iteration method.
There are still some interesting future works that need to be done. For example, we can extend Algorithm 2.1 for other structure TCP and discuss its convergence.

Founding
The work was supported by the Educational Commission of Guangdong Prov-

Conflicts of Interest
The author declares no conflicts of interest regarding the publication of this paper.