The relaxed asymmetric HSS-like iteration algorithms for a class of weakly nonlinear complementarity problems

. The problem of finding solutions to a class of weakly nonlinear complementarity problems is studied. And then, the authors present the relaxed asymmetric HSS-like iteration algorithm, the asymmetric HSS-like iteration algorithm and the HSS-like iteration algorithm for a class of weakly nonlinear complementarity problems. Under suitable conditions, they establish the convergence theory of the algorithms.


Introduction
This paper focuses on a class of weakly nonlinear complementarity problems, which is to find a pair of real vectors ‫ݎ‬ = ‫ݖܣ‬ + ‫)ݖ(݂‬ and ‫ݖ‬ ‫א‬ ܴ such that Where ‫ܣ‬ ‫א‬ ܴ × and ‫ݍ‬ ‫א‬ ܴ are given real matrix and vector, respectively, ‫:ܨ‬ ܴ ՜ ܴ is a given nonlinear mapping, usually arises from many scientific computing and engineering applications, for instance, the network equilibrium problem, the contact problem, image processing [1,2] and the free boundary problem [3].This kind of problems usually come from the numerical solution of some variational inequality problems with nonlinear source term.If ‫)ݖ(ܨ‬ is an affine function, the nonlinear complementarity problem (1) will reduce to the linear complementarity problem.If the linear part ‫ݔܣ‬ is stronger than the nonlinear part ‫)ݖ(ܨ‬ under a certain norm definition, equation ( 1) is called a weakly nonlinear complementarity problem.We denote its solution by ‫ݖ‬ ‫כ‬ .
where ߙ is a given nonnegative constant and ߚ is a given positive constant, and ‫ܪ‬ is the Hermitian part of ‫,ܣ‬ ܵ the skew-Hermitian part.
Step 4. Test for termination.If ‫ݖ‬ ାଵ satisfies a prescribed stopping rule, terminate.Otherwise, return to Step 2 with ݇ replaced by ݇ + 1.
In this paper, based on the HS splitting, we constructed the relaxed asymmetric HSSlike iteration algorithms for a class of weakly nonlinear complementarity problems.This paper is organized as follows.In the next section, we present some necessary notation and preliminary results.In Section 3 we describe the relaxed asymmetric HSS-like iteration algorithm.In Section 4, we analyze the convergence properties of the iteration algorithm under suitable conditions.

Preliminary
In this section we review some know results needed in section 4. To formulate them we begin with some basic notation used throughout the remaining part of this paper.

The relaxed asymmetric HSS-like iteration algorithm
In this section, we extend the nonlinear relaxed asymmetric HSS-like iteration method for a class of weakly nonlinear systems [7] to a class of weakly nonlinear complementarity problems.
Let ‫ܣ‬ = ‫ܪ‬ + ܵ be the Hermitian and skew-Hermitian splitting of A, where Then we can present the following extrapolated relaxed Hermitian and skew-Hermitian splitting iteration algorithm for a class of weakly nonlinear complementarity problems, which can be considered as an extension of extrapolated Hermitian and skew-Hermitian splitting iteration algorithm for linear equations proposed in [7].Algorithm 3.1 (The relaxed asymmetric HSS-like iteration algorithm) Step 1. Initialization.Let ‫ݖ‬ () ‫א‬ ܴ be any given initial value, set ݇ = 0.
Step 3. General iteration.Let ‫ݖ‬ (ାଵ) be an arbitrary solution of the following weakly nonlinear complementarity problems: find ‫ݖ‬ ‫א‬ ܴ , such that , ߚ is a given positive constant.
Step 4. Test for termination.If ‫ݖ‬ (ାଵ) satisfies a prescribed stopping rule, terminate.Otherwise, return to Step 2 with with ݇ replaced by ݇ + 1.

The convergence of relaxed asymmetric HSS-like iteration algorithmy
In this section, we establish the convergence theory for algorithm 3.1.Before the discussion of the convergence,we first introduce some useful lemmas.
Under this assumption, the inequality (13) holds clearly if ‫ݖ‬ (,ଵ) = 0, because the j-th component of the left hand vector in ( 8) is then nonpositive and the right-hand component is always nonnegative.
Applying above lemma to the relaxed asymmetric HSS-like iteration algorithm, we obtain the following convergence property.
where ‫)ܪ(ߣ‬ is the spectral set of ‫ܪ‬ and ߜ(ܵ) is the singular-value set of ܵ, Suppose that ܾܽ < 1.When Then, the iteration sequence ൛z (୩) ൟ generated by Algorithm 3.1 converges to the solution of the weakly nonlinear complementarity problem (1).