数学与统计学院知识库

互补问题是指在一定的空间内找到一对非负变量使其满足一种互补关系, 而这种关系反应了一种广泛存在的基本关系. 自从上世纪 60 年代互补问题被引入和研究以来, 就一直受到众多数学研究者和数学爱好者的广泛关注. 互补问题是数学领域的一个热门课题, 关于它的求解算法也不断得到更新. 互补问题与对策论, 规划问题, 変分学, 经济学, 力学等学科有紧密联系, 在科学计算和工程应用中也有着广泛的应用.

本文旨在考虑研究一类非线性互补问题的有效算法. 首先将互补问题转化为等价的隐式不动点方程. 然后, 基于该不动点方程我们建立了模系同步多分裂 (MSM) 迭代算法. 另外, 我们还介绍了几种不同的 MSM 方法, 包括 Jacobi, Gauss-Seidel, 连续超松弛 (SOR), 以及加速超松弛 (AOR) 算法. 同时给出了系数矩阵  A 是 H₊ 矩阵时迭代方法的收敛理论. 文章最后利用数值实验验证了这些算法的可行性和有效性.

Complementary problem is defined as to find a pair of nonnegative variables in a certain space to make it meet a complementary relationship, and the relationship reflects a basic relations of the widespread. Since 60s of the last century, the complementary problem was introduced and researched, it has received extensive attention of numerous mathematical researchers and enthusiasts. Complementary problems are not only a hot topic in the field of mathematics, but their algorithms are also constantly updated. They have close connection with game theory, programming, economics, mechanics disciplines; while there is a wide range of applications in scientific computing and engineering applications.

This paper aims to consider an effective algorithm for a restricted class of nonlinear complementarity problems.
It is first reformulated into an equivalent implicit fixed-point equation in this work. Then we establish a modulus-based synchronous multisplitting iteration method based on the fixed-point equation. Moreover, several kinds of special choices of the iteration methods including multisplitting relaxation methods such as Jacobi, Gauss-Seidel, successive overrelaxation (SOR), and accelerated overrelaxation (AOR) of the modulus type are presented. Convergence theorems for these iteration methods are proven when the system matrix A is an H₊ matrix. Numerical results are also provided to confirm the efficiency of these methods in actual implementations.