In this paper, we propose the horizontal implicit complementarity problems, which cover many complementarity problems that have wide applications. After reformulating the horizontal implicit complementarity problem as an implicit fixed-point equation, the modulus-based matrix splitting iteration methods are applied to solve the problems. Besides, this paper introduces a new kind of matrix splitting, called modulus-based Hermitian and skew-Hermitian matrix relaxation splitting. The corresponding modulus-based matrix splitting iteration method is shown superior to the nonsmooth Newton's method and the existing matrix splitting schemes by some numerical experiments. The convergence analysis is given and validates that our methods are more practical in applications.

在本文中，我们提出了水平隐式互补问题，涵盖了许多具有广泛应用的互补问题。将水平隐式互补问题重新表述为隐式定点方程后，采用基于模的矩阵分裂迭代方法来求解该问题。此外，本文还介绍了一种新的矩阵分裂，称为基于模的埃尔米特和斜埃尔米特矩阵松弛分裂。数值实验表明相应的基于模的矩阵分裂迭代方法优于非光滑牛顿法和现有的矩阵分裂方案。给出了收敛性分析，验证了我们的方法在应用中更实用。