求解一类非线性互补问题的广义模基矩阵分裂迭代法

通过引入新的正对角参数矩阵, 提出了求解$H$-矩阵非线性互补问题的广义模基矩阵分裂迭代法和广义二步模基矩阵分裂迭代法, 取定特殊的正对角参数矩阵和矩阵分裂后, 两种算法都可转化为已有的模基矩阵分裂迭代法, 因此是已有求解线性互补问题和非线性互补问题模基矩阵分裂迭代法的推广. 利用$H$-矩阵的相关性质建立了两种算法的收敛性分析, 在算法收敛的充分条件中, $H$-分裂的假设比已有的非线性互补问题模基矩阵分裂迭代法$H$-相容分裂的收敛条件更弱; 另外, 所得到的正对角参数矩阵的收敛域比已有非线性互补问题模基矩阵分裂迭代法的收敛域更大, 因此收敛性结果是已有算法收敛性结果的推广改进, 这表明新的正对角参数矩阵是有效的.

A general modulus-based matrix splitting iteration method for a class of nonlinear complementarity problems

By introducing a new positive diagonal parameter matrix, a general modulus-based matrix splitting iteration method and a general two-steps modulus-based matrix splitting iteration method for solving nonlinear complementarity problem of $H$-matrices are proposed. The two methods can reduce to the existing modulus-based matrix splitting iteration method by choosing special positive diagonal parameter matrices and splitting. Hence they generalize the existing modulus-based matrix splitting iteration method for solving linear and nonlinear complementarity problems. The convergence analysis of the two methods are given by the properties of $H$-matrix. The $H$-splitting assumption in the sufficient condition of convergence is weaker than the $H$-compatible splitting assumption of the existing modulus-based matrix splitting iteration method for nonlinear complementarity problem; In the other hand, the convergence domain of the positive diagonal parameter matrix is larger than that in the existing modulus-based matrix splitting iteration method for nonlinear complementarity problem. So the convergence results of the methods improve the existing ones, which shows the significance of introducing the new positive diagonal parameter matrix.