A Tensor is a multi-arrray. We denote by the space of all m-order and n-dimension tensors. is in the form of
If the entries are invariant under any permutation of their indices, then is called a symmetric tensor. We denote by the space of all symmetric tensors in .
In this paper, we discuss the following tensor complementarity problem (simplified as TCP ): finding a point such that
where , and . is an n-dimensional vector, whose ith component is given by 
is an matrix, whose ith row and jth column is given by
It is obvious that .
Tensor complementarity problem has many applications, such as nonlinear compressed sensing, communications, DNA microarrays, n-person noncooperative game and so on, see for example  . TCP has received much attention and has taken good progress in recent years  - , such as the structure of the solution set, the global uniqueness solvability and the error bound and so on. Bai, Huang and Wang  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  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  proposed an iterative method to find the sparsest solutions to theZ-tensor complementarity problem with a non-positive constant term. Xu, Li and Xie  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 semismooth Newton method for TCP, under the assumption that is a concave function, we establish the monotone convergence theorem for the proposed method. Here is concave means is a concave function, .
For convenience of presentation, we introduce some concepts and notations which will be used throughout the paper. We denote . Let with . Denote by the r-dimensional subvector of and its elements are , . Similarly, we denote as the r-dimensional subvector of q and its elements are , .
2. Semismooth Newton Method and Its Convergence
We first introduce a new class of structure tensor, called M-like tensor.
Definition 2.1. A tensor is called an M-like tensor, if for , is an M-matrix.
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.
Example 2.1. Let be a 4th-order 2-dimensional tensor with elements
It is easy to verify is an M-like tensor over . In the following, without specification, we always suppose is an M-like tensor.
Let be the well-known Fischer-Burmeister function defined by
It is easy to see that TCP (1.1) can be transformed to the system of nonsmooth equations:
Let denote the B-subdifferential of H at . , if it satisfies
For any can be expressed as follows.
It is easy to see that and for .
Now, we present semismooth Newton method for TCP (1.1).
Algorithm 2.1 (Semismooth Newton Method)
Step 0. Choose and initial point . Let .
Step 1. If , stop.
Step 2. Choose , calculate satisfying the following linear equations
Step 3. Calculate
Let and goto Step 1.
Theorem 2.2.  Let be the solution of (1). Then the sequence generated by (6) is Q-quadratically convergent to if is sufficiently close to .
In the following, we will prove the sequence monotonically converges to the solution of (1). First, we give some useful lemmas.
Lemma 2.3.  Let be an M-matrix and satisfy for . Then B is an M-matrix and
Furthermore, any principle submatrix of A is again an M-matrix.
By the definition of V, the ith row of V can be presented by
where is the ith row of and is the ith row of the identity matrix. It is easy to verify , where with
where is the ith diagonal element of . Noting that with the off-diagonal element being nonpositive, we have B is an M-matrix and by Lemma 2.3. This together with K being a positive diagonal matrix, we obtain the following lemma.
Lemma 2.4. Let the matrix V be defined by (2.3), then V is an M-matrix.
Define the set D by
As p always belongs to D, D is not empty.
Lemma 2.5. Let be the solution of (1.1). Then and
Proof. Since is the solution of (1.1), we have and . This implies that .
For any , we have and . Associated to x, we define two disjoint index sets as follows:
This together with the fact implies for all . Since , we have
Noting that is an M-like tensor, by Theorem 4 in , we have is a strong T-monotone function. Hence by Lemma 2.2 in , we immediately have that . This completes the proof.
Lemma 2.6. Let . Then the sequence generated by (2.6) is contained in D and satisfies
Proof. We prove the conclusion by induction. For simplicity, we denote and by x and y respectively. Suppose , we only need to verify and . Since , by (2.5) we have
Hence, we get
Now, we prove . Associated to x, we define four disjoint index sets as follows:
Then by (2.1), (2.2), (2.4), we have
If , by (2.3), (2.5), (2.8) we get and . This together with (2.2), (2.1) implies .
If , . Since is concave, by (2.3), (2.5), (2.8) again, we have
If , by (2.5) and (2.3), we obtain
where and . Hence, we have
If , then , which implies .
If , then by (2.4) we have , and then . Since is an M-like tensor, and for , we have
Similarly, if , by simple calculations,
Therefore, we have
The above argument has shown . From (2.7), we have and . The proof is completed.
Noting that the sequence is uniformly bounded, by Lemmas 2.5 and 2.6 and Theorem 2.2, it is easy to obtain the following theorem.
Theorem 2.7. Let and be the sequence generated by (2.6). Then we have for all
Moreover, the sequence converges to Q-quadratically.
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.
The work was supported by the Educational Commission of Guangdong Province, China (grant No. 2019KTSCX172).
 Huang, Z. and Qi, L. (2017) Formulating an n-Person Noncooperative Game as a Tensor Complementarity Problem. Computation Optimization and Applications, 66, 557-576.
 Bai, X.L., Huang, Z.H. and Wang, Y. (2016) Global Uniqueness and Solvability for Tensor Complementarity Problems. Journal of Optimization Theory and Applications, 170, 72-84.
 Che, M., Qi, L. and Wei, Y. (2016) Positive-Definite Tensors to Nonlinear Complementarity Problems. Journal of Optimization Theory and Applications, 168, 475-487.
 Wang, Y., Huang, Z.H. and Bai, X.L. (2016) Exceptionally Regular Tensors and Tensor Complementarity Problems. Optimization Methods and Software, 31, 815-828.
 Xu, H.R., Li, D.H. and Xie, S.L. (2019) An Equivalent Tensor Equation to the Tensor Complementarity Problem with Positive Semi-Definite Z-Tensor. Optimization Letters, 13, 685-694.