As a generalization of the well-known linear complementarity problem, tensor complementarity problem (TCP) has been studied extensively in the literature from theoretical perspective. In this paper, we consider a class of TCPs equipped with nonsingular (not necessarily symmetric) $\mathcal{M}$-tensors. The considered TCPs can be regarded as a special class of nonlinear complementarity problems (NCPs), but the underlying mappings in TCPs are not necessarily $P_0$-functions, which preclude the possibility of applying existing Newton-type methods for NCPs to TCPs directly. By introducing a pivot minimum function, we propose an index detecting algorithm, which efficiently exploits the beneficial properties of nonsingular $\mathcal{M}$-tensors and goes beyond algorithmic frameworks designed for general NCPs. Moreover, the proposed algorithm is well-defined in the sense that it generates a nonnegative element-wise nonincreasing sequence converging to a solution of the problem under consideration. Finally, numerical results further support the efficiency and reliability of the proposed algorithm.

作为众所周知的线性互补问题的推广，张量互补问题（TCP）已从理论的角度在文献中进行了广泛的研究。在本文中，我们考虑一类配备非单数（不一定是对称的）的TCP$\mathcal{中号}$张量。可以将考虑的TCP视为一类特殊的非线性互补问题（NCP），但TCP中的基础映射不一定$P_0$功能，从而排除了将现有的用于NCP的Newton型方法直接应用于TCP的可能性。通过引入枢轴极小值函数，我们提出了一种索引检测算法，该算法可有效利用非奇异函数的有益特性。$\mathcal{中号}$-张量，并超越了为通用NCP设计的算法框架。此外，从算法生成非负元素逐级非递增序列收敛到所考虑问题的解决方案的意义上来说，该算法是定义明确的。最后，数值结果进一步证明了所提算法的有效性和可靠性。