Convergence analysis of modulus-based matrix splitting iterative methods for implicit complementarity problems

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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$H_{+}$\end{document}H+-matrix, respectively.


Introduction
Consider the following implicit complementarity problem [2], abbreviated ICP, of finding a solution u ∈ R n to um(u) ≥ 0, w := Au + q ≥ 0, um(u) T w = 0, (1.1) where A = (a ij ) ∈ R n×n , q = (q 1 , q 2 , . . . , q n ) T ∈ R n , and m(·) stands for a point-to-point mapping from R n into itself. We further assume that u-m(u) is invertible. Here (·) T denotes the transpose of the corresponding vector. In the fields of scientific computing and economic applications, many problems can result in the solution of the ICP (1.1); see [3,4]. In [2], the authors have shown how all kinds of complementarity problems can be transformed into the ICP (1.1). In the same paper, the authors have studied sufficient conditions of the existence and uniqueness of solution to the ICP (1.1). In particular, if the point-to-point mapping m is a zero mapping, then the ICP (1.1) is equivalent to u ≥ 0, w := Au + q ≥ 0, u T w = 0, (1.2) which is known as the linear complementarity problem (abbreviated LCP) [5].
In the past few decades, much more attention has been paid to find efficient iterative methods for solving the ICP (1.1). Based on a certain implicitly defined mapping F and the idea of iterative methods for solving LCP (1.2), Pang proposed a basic iterative method where u (0) is a given initial vector, and established the convergence theory. For more discussions on the mapping F and its role in the study of the ICP (1.1), see [6]. By changing variables, Noor equivalently reformulated the ICP (1.1) as a fixed-point problem, which can be solved by some unified and general iteration methods [7]. Under some suitable conditions, Zhan et al. [8] proposed a Schwarz method for solving the ICP (1.1). By reformulating the ICP (1.1) into an optimization problem, Yuan and Yin [9] proposed some variants of the Newton method.
Recently, the modulus-based iteration methods [10], which were first proposed for solving the LCP (1.2), have attracted attention of many researchers due to their promising performance and elegant mathematical properties. The basic idea of the modulus iteration method is transforming the LCP into an implicit fixed-point equation (i.e., the absolute equation [11]). 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. Bai [13] presented a class of modulus-based matrix splitting (MMS) iteration methods, which inherit the merits of the modulus iteration method. Some general cases of the MMS methods have been studied in [14][15][16][17][18]. Hong and Li extended the MMS methods to solve the ICP (1.1). Numerical results showed that the MMS iteration methods are more efficient than the well-known Newton method and the classical projection fixed-point iteration methods [1]. In this paper, we further consider the iteration scheme of the MMS iteration method and will demonstrate a complete version about the convergence theory of the MMS iteration methods. New convergence conditions are presented when the system matrix is a positive-definite matrix and an H + -matrix, respectively.
The outline of this paper is as follows. In Section 2, we give some preliminaries. In Section 3, we introduce the MMS iteration methods for solving the ICP (1.1). We give a complete version of convergence analysis of the MMS iteration methods in Section 4. Finally, we end this paper with some conclusions in Section 5.

Preliminaries
In this section, we recall some useful notations, definitions, and lemmas, which will be used in analyzing the convergence of the MMS iteration method for solving the ICP (1.1). Let be two matrices. If their elements satisfy a ij ≥ b ij (a ij > b ij ), then we say that A ≥ B (A > B). If a ij ≥ 0 (a ij > 0), then A = (a ij ) ∈ R m×n is said to be a nonnegative (positive) matrix. If a ij ≤ 0 for any i = j, then A is called a Z-matrix. Furthermore, if A is a Z-matrix and A -1 ≥ 0, then A is called an M-matrix. A matrix A = ( a ij ) ∈ R n×n is called the comparison matrix of a matrix A if the elements a ij satisfy i, j = 1, 2, . . . , n.
A matrix A is called an H-matrix if its comparison matrix A is an M-matrix, and an H +matrix if it is an H-matrix and its diagonal entries are positive; see [19]. A matrix A is called a symmetric positive-definite if A is symmetric and satisfies x T Ax > 0 for all x ∈ R n \ {0}.
In addition, A = F -G is said to be a splitting of the matrix A if F is a nonsingular matrix, and an H-compatible splitting if it satisfies A = F -|G|. We use |A| = (|a ij |) and A 2 to denote the absolute and Euclidean norms of a matrix A, respectively. These symbols are easily generalized to the vectors in R n ; σ (A), ρ(A), and diag(A) represent the spectrum, spectral radius, and diagonal part of a matrix A, respectively.

Modulus-based matrix splitting iteration methods for ICP
To present the MMS iteration method, we first give a lemma that shows that the ICP (1.1) is equivalent to a fixed-point equation. , then x = γ 2 (u --1 wm(u)) satisfies the implicit fixed-point equation is a solution of the ICP (1.1).
Then based on the implicit fixed-point equation (3.1), Hong and Li [1] established the following MMS iteration methods for solving the ICP (1.1).
Method 3.1 converges to the unique solution of ICP (1.1) under mild conditions and has a faster convergence rate than the classical projection fixed-point iteration methods and the Newton method [1]. However, Method 3.1 cannot be directly applied to solve the ICP (1.1). On one hand, the authors did not specify how to solve w (k) . On the other hand, step 2(2) is actually an inner iteration at the kth outer iteration. The outer iteration information should be presented in the MMS iteration method. To better show how the MMS iteration method works, we give a complete version as follows.
From Method 3.1 or Method 3.2 we can see that the MMS iteration method belongs to a class of inner-outer iteration methods. In general, the convergence rate of inner iteration has great effect on total steps of the outer iteration. However, in actual implementations the inner iterations need not communicate. Note that the number of outer iterations decreases as the number of inner iteration increases. This may lead to the reduction of the total computing time, provided that the decrement of communication time is not less than the increment of computation time for the inner iterations. So, a suitable choice of the number of inner iterations is very important and can greatly improve the computing time for solving the ICP (1.1). To efficiently implement the MMS iteration method, we can fix the number of inner iterations or choose a stopping criterion about residuals of inner iterations at each outer iteration. For the inner iteration implementation aspects of the modulus-based iteration method, we refer to [12,23] for details.

Convergence analysis
In this section, we establish the convergence theory for Method 3.2 when A ∈ R n×n is a positive-definite matrix and an H + -matrix, respectively.
To this end, we first assume that there exists a nonnegative matrix N ∈ R n×n such that In addition, from Method 3.2 we have where w ( * ) = Au ( * ) + q. Subtracting (4.1) from (3.5) and taking absolute values on both sides, we obtain Similarly, subtracting (4.2) from (3.4), we have where where δ 3 = 2 j i=0 δ i 1 δ 2 + I. Similarly, from (3.3) and (4.3) we have Finally, substituting (4.7) into (4.6), we get Therefore, if ρ(Z) < 1, then Method 3.2 converges to the unique solution of the ICP (1.1). We summarize our discussion in the following theorem. In Theorem 4.1 a general sufficient condition is given to guarantee the convergence of the MMS iteration method. However, this condition may be useless for practical computations. In the following two subsections, some specific conditions are given when the system matrix A is positive-definite and an H + -matrix, respectively.

Remark 4.1
Although in [1] the modulus-based iteration method was proposed based on a matrix splitting, the authors just considered the following iteration scheme in analyzing the convergence that is, the convergence of the modulus-based matrix splitting iteration method was not actually proved in [1]. In Theorems 4.1 and 4.2, we give a complete version of the convergence of the MMS iteration method (i.e., Method 3.2). These results generalize those in [1, Theorems 4.1 and 4.2].

Conclusions
In this paper, we have studied a class of modulus-based matrix splitting (MMS) iteration methods proposed in [1] for solving implicit complementarity problem (1.1). We have modified implementation of the MMS iteration method. In addition, we have demonstrated a complete version of the convergence theory of the MMS iteration method. We have obtained new convergence results when the system matrix A is a positive-definite matrix and an H + -matrix, respectively.