We are interested in finding a solution to the tensor complementarity problem with a strong M-tensor, which we call the M-tensor complementarity problem. We propose a lower dimensional linear equation approach to solve that problem. At each iteration, only a lower dimensional system of linear equation needs to be solved. The coefficient matrices of the lower dimensional linear systems are independent of the iteration after finitely many iterations. We show that starting from zero or some nonnegative point, the method generates a sequence of iterates that converges to a solution of the problem monotonically. We then make an improvement to the method and establish its monotone convergence. At last, we do numerical experiments to test the proposed methods. The results positively support the proposed methods.

我们感兴趣的是找到具有强M张量的张量互补问题的解决方案，我们称其为M张量互补问题。我们提出了一种低维线性方程方法来解决该问题。在每次迭代中，只需要求解一个低维线性方程组。在有限次迭代之后，低维线性系统的系数矩阵与迭代无关。我们表明，从零或某个非负点开始，该方法生成一个迭代序列，这些序列单调收敛到问题的解决方案。然后，我们对该方法进行了改进，并建立了其单调收敛性。最后，我们进行了数值实验以验证所提出的方法。结果肯定地支持了所提出的方法。