Modulus-Based Multisplitting Iteration Method for a Class of Weakly Nonlinear Complementarity Problem

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

“…Example 4. 4 In this example, we consider Corollary 3.3 and the MSOR iteration method for solving (1), that is α = β in (24). According to (29) mentioned at the end of Section 3, we test the convergence and the numerical results for this method.…”

“…In this paper, we consider the nonlinear complementarity problem (NCP(A, q, ϕ)), which is to find z ∈ R n such that z T w = 0 with z ≥ 0, w = Az + q + ϕ(z) ≥ 0, (1) where A = (a ij ) ∈ R n×n , q ∈ R n , and the nonlinear term ϕ(z) = (ϕ 1 (z 1 ), ϕ 2 (z 2 ), • • • , ϕ n (z n )) T satisfies | dϕ i dz i | ≤ ji for i = 1, 2, ..., n. In recent decades, various types of nonlinear complementarity problems have been studied by many researchers, see [1][2][3][4][5][6] and the references therein.…”

“…After that, a series of modulus-based type matrix splitting iteration methods have been proposed, for instance, the general modulus-based matrix splitting method [14], the preconditioned modulus-based matrix splitting iteration method [15], the two-step modulus-based matrix splitting iteration method [16], the accelerated modulus-based matrix splitting iteration methods [17], and so on. Since these modulus-based type matrix splitting iteration methods are usually very practical and effective, many authors extended this kind of methods to solve other complementarity problems, such as the implicit complementarity problems [6,18,19], the weakly nonlinear complementarity problems [3,4], the horizontal nonlinear complementarity problems [20,21], the quasi-complementarity problems [22], and so on.…”

Improved Inexact Alternating Direction Methods for a Class of Nonlinear Complementarity Problems

“…The convergence results have been then extended from H-compatible splitting to H-splitting of NCPs with H + -matrices [30]. Extensions of MMS methods, such as modified MMS method [33], accelerated MMS method [29], relaxed MMS method [58], two-step MMS method [46,55], preconditioned MMS method [54,59] and modulus-based matrix multisplitting methods [43,44,48,49], developed in recent years improved the original one. Besides, a modulus-based nonsmooth Newton method and sign-based methods are studied in [57] and [56], respectively.…”