Abstract
In this paper, we demonstrate a complete version of the convergence theory of the modulus-based matrix splitting iteration methods for solving a class of implicit complementarity problems proposed by Hong and Li (Numer. Linear Algebra Appl. 23:629-641, 2016). New convergence conditions are presented when the system matrix is a positive-definite matrix and an H_{+}-matrix, respectively.
Highlights
Consider the following implicit complementarity problem [2], abbreviated ICP, of finding a solution u ∈ Rn to u – m(u) ≥ 0, w := Au + q ≥ 0, u – m(u) T w = 0, (1.1)where A = ∈ Rn×n, q = (q1, q2, . . . , qn)T ∈ Rn, and m(·) stands for a point-to-point mapping from Rn into itself
We recall some useful notations, definitions, and lemmas, which will be used in analyzing the convergence of the modulus-based matrix splitting (MMS) iteration method for solving the ICP (1.1)
We summarize our discussion in the following theorem
Summary
Wang et al Journal of Inequalities and Applications (2018) 2018:2 idea of iterative methods for solving LCP (1.2), Pang proposed a basic iterative method u(k+1) = F u(k) , k ≥ 0, where u(0) is a given initial vector, and established the convergence theory. Noor equivalently reformulated the ICP (1.1) as a fixed-point problem, which can be solved by some unified and general iteration methods [7]. To accelerate the convergence rate of the modulus iteration method, Dong and Jiang [12] introduced a parameter and proposed a modified modulus iteration method They showed that the modified modulus iteration method is convergent unconditionally for solving the LCP when the system matrix A is positive-definite.
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