Relaxed Modulus-Based Matrix Splitting Methods for the Linear Complementarity Problem †

: In this paper, we obtain a new equivalent ﬁxed-point form of the linear complementarity problem by introducing a relaxed matrix and establish a class of relaxed modulus-based matrix splitting iteration methods for solving the linear complementarity problem. Some sufﬁcient conditions for guaranteeing the convergence of relaxed modulus-based matrix splitting iteration methods are presented. Numerical examples are offered to show the efﬁcacy of the proposed methods.


Introduction
In this paper, we focus on the iterative solution of the linear complementarity problem, abbreviated as 'LCP(q, A)', whose form is w = Az + q ≥ 0, z ≥ 0 and z T w = 0, (1) where A ∈ R n×n and q ∈ R n are given, and z ∈ R n is unknown, and for two s × t matrices G = (g ij ) and H = (h ij ) the order G ≥ (>)H means g ij ≥ (>)h ij for any i and j. As is known, as a very useful tool in many fields, such as the free boundary problem, the contact problem, option pricing problem, nonnegative constrained least squares problems, see [1][2][3][4][5], LCP(q, A) is most striking in the articles, see [6][7][8][9][10][11].
Designing iteration methods to fast and economically computing the numerical solution of the LCP(q, A) is one of the hotspot nowadays, which were widely discussed in the articles, see [1,2,8,9,12,13] for more details. Recently, by using w = 1 γ Ω(|x| − x) and z = 1 γ (|x| + x) with γ > 0 for the LCP(q, A), Bai in [14] expressed the LCP(q, A) as the fixed-point form where A = M − N is a splitting of matrix A and Ω denotes a positive diagonal matrix, and then first desgined a class of modulus-based matrix splitting (MMS) iteration methods.
In this paper, based on the MMS method, we will create a type of new iteration methods to solve the LCP(q, A). Our strategy is to introduce a relaxed matrix for both sides of Equation (2) and obtain a new equivalent fixed-point form of the LCP(q, A). Based on this new equivalent form, we can establish a class of relaxed modulus-based matrix splitting (RMMS) iteration methods for solving the LCP(q, A). This class of new iteration methods not alone inherits the virtues of the modulus-based methods. A more important one is that we can choose the relaxed matrix to enhance the computational efficiency of the classical MMS method in [14]. To guarantee the convergence of the RMMS iteration method, some sufficient conditions will be given under suitable conditions. Numerical examples are provided to verify that the RMMS iteration method are feasible and overmatch the classical MMS iteration method in terms of the computational efficiency.
The layout of this paper is as follows. In Section 2, for the sake of discussion in the rest of this paper, we provide some necessary definitions, lemmas and notations. In Section 3, we present a class of relaxed modulus-based matrix splitting (RMMS) iteration methods to solve the LCP(q, A). The convergence conditions of the RMMS iteration method are presented in Section 4. Numerical experiments with regard to the proposed methods compared to the classical MMS iteration method are reported in Section 5. In Section 6, we summarize this work. Finally, some simple discussions are given in Section 7.

Preliminaries
Some necessary definitions, lemmas and notations, which are used in the sequel discussions, are primitively introduced in this section. Let all of its principal minors are positive [33,34]. In addition, we let |A| = (|a ij |).
Let A = M − N be a splitting of matrix A ∈ R n×n . Then it is named as an M-splitting if M is a nonsingular M-matrix and N ≥ 0; an H-splitting if M − |N| is a nonsingular M-matrix. As is known, if A = M − N is an M-splitting and A is a nonsingular M-matrix, then ρ(M −1 N) < 1, where ρ(·) indicates the spectral radius (the maximum value of the absolute value of the eigenvalues) of the matrix, see [33,34]. Finally, · 2 denotes the Euclidean norm on R n . Lemma 1 ([19]). Let A = (a ij ) ∈ R n×n with a ij ≥ 0. If there exists u ∈ R n with u > 0 such that Au < u, then ρ(A) < 1.

Lemma 3 ([34]). Let A be an M-matrix and B be a Z-matrix with A ≤ B. Then B is an M-matrix.
In addition, there exists a famous result for the existence and uniqueness of the LCP(q, A), that is to say, the LCP(q, A) has a unique solution if and only if a matrix A is a P-matrix, see [1]. Obviously, when A is an H + -matrix, the LCP(q, A) has a unique solution as well.

Relaxed Modulus-Based Matrix Splitting Method
In this section, we introduce a class of relaxed modulus-based matrix splitting (RMMS) iteration methods for solving the LCP(q, A). For this purpose, by introducing an identical equation Rx − Rx = 0 for Equation (2), where R ∈ R n×n is a given relaxed matrix, then we obtain or, Based on Equations (3) and (4), we can establish the following iteration method, which is named as a class of relaxed modulus-based matrix splitting (RMMS) iteration methods for the LCP(q, A).

Method 1.
Let A = M − N be a splitting of the matrix A ∈ R n×n . Let γ > 0 and matrix Ω + M − R be nonsingular, where Ω is a positive diagonal matrix and R ∈ R n×n is a given relaxed matrix. Given an initial vector where x (k+1) can be obtained by solving the linear system Clearly, when R = 0, method 1 reduces to the well-known MMS iteration method in [14]. Similarly, by introducing an identical equation, Wu and Li in [22] presented twosweep modulus-based matrix splitting iteration methods for the LCP(q, A). The goal of introducing the relaxed matrix R in (5) or (6) is that the computational efficiency of the RMMS iteration method may be better than the classical MMS iteration method in [14].
In fact, method 1 is a general framework of RMMS iteration methods for solving the LCP(q, A). This implies that we can construct some concrete forms of RMMS iteration methods by the specific splitting matrix of matrix A and the iteration parameters. If we take where D, L and U, respectively, are the diagonal, the strictly lower-triangular and the strictly upper-triangular parts of the matrix A, then this leads to the relaxed modulusbased AOR (RMAOR) iteration method with z (k+1) = 1 γ (|x (k+1) | + x (k+1) ). When α = β, α = β = 1, and α = 1, β = 0, respectively, the RMAOR method (7) can yield the corresponding relaxed modulus-based SOR (RM-SOR) method, the relaxed modulus-based Gauss-Seidel (RMGS) method and the relaxed modulus-based Jacobi (RMJ) method.

Convergence Analysis
In this section, some sufficient conditions are given to guarantee the convergence of method 1. Theorem 1. Let A = M − N be a splitting of the matrix A ∈ R n×n with A being a P-matrix, and matrix Ω + M − R be nonsingular, where Ω is a positive diagonal matrix and R ∈ R n×n is a given relaxed matrix. Let When δ(R) < 1, Method 1 with γ > 0 converges to the unique solution z * ∈ R n + of the LCP(q, A) for an initial vector.
Proof. Let (z * , w * ) be a solution pair of the LCP(q, A). Then Based on (6) and (8), note that matrix Ω + M − R is nonsingular, we can obtain This indicates that Obviously, when δ(R) < 1, method 1 is convergent.
the following corollary can be obtained.

Corollary 1.
Let A = M − N be a splitting of the matrix A ∈ R n×n with A being a P-matrix, and matrix Ω + M − R be nonsingular, where Ω is a positive diagonal matrix and R ∈ R n×n is a given relaxed matrix. Letδ Whenδ(R) < 1, Method 1 with γ > 0 converges to the unique solution z * ∈ R n + of the LCP(q, A) for an initial vector.
Similar to the above proof, if we take | · | instead of · 2 , we can also obtain Corollary 2.

Corollary 2.
Let A = M − N be a splitting of the matrix A ∈ R n×n with A being a P-matrix, and matrix Ω + M − R be nonsingular, where Ω is a positive diagonal matrix and R ∈ R n×n is a given relaxed matrix. Let When ρ(Φ) < 1 or ρ(Ψ) < 1, Method 1 with γ > 0 converges to the unique solution z * ∈ R n + of the LCP(q, A) for an initial vector.

Theorem 2.
Let A = M − N be a splitting of the matrix A ∈ R n×n with A being a P-matrix, and matrix Ω + M be nonsingular and matrix I − |(Ω + M) −1 R| be nonsingular M-matrix, where Ω is a positive diagonal matrix and R ∈ R n×n is a given relaxed matrix. Let When ρ(Θ) < 1, method 1 with γ > 0 converges to the unique solution z * ∈ R n + of the LCP(q, A) for an initial vector.

Corollary 3.
Let A = M − N be a splitting of the matrix A ∈ R n×n with A being a P-matrix, and matrix Ω + M be nonsingular and I − |(Ω + M) −1 R| be nonsingular M-matrix, where Ω is a positive diagonal matrix and R ∈ R n×n is a given relaxed matrix. Let When ρ(Θ) < 1, method 1 with γ > 0 converges to the unique solution z * ∈ R n + of the LCP(q, A) for an initial vector.
. Then method 1 with γ > 0 converges to the unique solution z * ∈ R n + of the LCP(q, A) for an initial vector.
Proof. First, we prove that Ω + M − R is an M-matrix. In fact, since In addition, where matrix ∆ = (δ ij ) ∈ R n×n satisfy It noted that By calculation, it is easy to obtain that matrix ∆ − |Ω − A| is a nonnegative diagonal matrix. It follows that matrix is an M-matrix. Further, there exists a positive vector u such that Therefore, Based on Lemma 1, we can obtain that ρ(W) < 1. which completes the proof.
When matrix R is a diagonal matrix, Corollary 4 can be obtained. Let matrix Ω = (ω ij ) ∈ R n×n satisfy ω ii ≥ a ii and 0 ≤ ω ij ≤ 1 2 (|m ij | + |n ij | − |a ij |). Then Method 1 with γ > 0 converges to the unique solution z * ∈ R n + of the LCP(q, A) for an initial vector.
When matrix R is a zero matrix, Corollary 4 reduces to the following result, which is a main result in [36].

Corollary 5. [36]
Let A = M − N be a splitting of the matrix A = (a ij ) ∈ R n×n , where A is an H + -matrix, and M − |N| be an M-matrix. Let Ω = (ω ij ) ∈ R n×n satisfy ω ii ≥ a ii and 0 ≤ ω ij ≤ 1 2 (|m ij | + |n ij | − |a ij |). Then method 1 with R = 0 and γ > 0 converges to the unique solution z * ∈ R n + of the LCP(q, A) for an initial vector.
Further, when matrix R is a zero matrix and Ω = (ω ij ) ∈ R n×n is a positive diagonal matrix, Corollary 4 reduces to the following result, which is a main result in [37].

Corollary 6.
[37] Let A = M − N be a splitting of the matrix A = (a ij ) ∈ R n×n , where A is an H + -matrix, and M − |N| be an M-matrix. Let the positive diagonal matrix Ω = (ω ij ) ∈ R n×n satisfy ω ii ≥ a ii . Then Method 1 with R = 0 and γ > 0 converges to the unique solution z * ∈ R n + of the LCP(q, A) for an initial vector.
When matrix R in Theorem 4 is a nonpositive diagonal matrix, Theorem 4 reduces to the following result.

Numerical Experiments
In this section, we utilize two examples to illustrate the computational efficiency of the RMMS iteration method in terms of iteration steps (IT) and elapsed CPU time (CPU) in seconds, and the following norm of absolute residual vectors (RES) where the minimum is componentwisely taken.
To show the advantages of the RMMS iteration method, we compare the RMMS iteration method with the classical MMS iteration method. During these tests, all initial vectors are chosen to be all the iterations are stopped once RES(z (k) )≤ 10 −5 or the number of iteration exceeds 500. For convenience, here we consider the relaxed modulus-based SOR (RMSOR) method and the modulus-based SOR (MSOR) method. The basis of this comparison is that the modulusbased SOR (MSOR) method in [14] outperforms other forms of the modulus-based matrix splitting iteration method, the projected relaxation methods and the modified modulus method. In actual implementations, we take Ω = D and γ = 2 for the RMSOR method and the MSOR method. All of computations are performed in MATLAB 7.0. In addition, in the following tables, '-' denotes that the iteration steps exceed 500 or the residual norms exceed 10 −5 .
In our computations, we take the following two examples, which were considered in [14,18]. Parts of two examples are symmetry, see case 1 of Example 1 and the diagonal matrix of Example 2. Of course, we consider the non-symmetry case, see case 2 of Example 1 and Example 2.
Example 2 (( [18]). Let the LCP(q, A) be given by In Examples 1 and 2, the value of m is chosen to be a prescribed positive integer, and then n = m 2 . For Example 1, we consider two cases: one is the symmetric case and the other is the nonsymmetric case. For the former, we take r = t = 1; for the latter, we take r = 1.5 and t = 0.5. In the implementations, the value of the iteration parameter α used in both the RMSOR method and the MSOR method is chosen to be 1.3. For convenience, the value of matrix is chosen to be R = 2I for the RMSOR method, where I is the identity matrix.
For Example 1, for different problem sizes of m and the different values of µ, the numerical results (including IT, CPU and RES) for the RMSOR method and the MSOR method are listed in Table 1 when r = t = 1. Clearly, the RMSOR method and the MSOR method can rapidly compute a satisfactory approximation to the solution of the LCP(q, A).
From the numerical results in Table 1, fixed the value of µ, the iteration steps and the CPU times of the RMSOR method and the MSOR method are incremental when the problem size n = m 2 is increasing. Whereas, fixing the value of the problem size n = m 2 , the iteration steps and the CPU times of the RMSOR method and the MSOR method are descended when the value of µ is increasing. This implies that both may be fit for the larger µ when as a solver for solving the LCP(q, A).
Base on the presented numerical results in Table 1, our numerical experiment show that the RMSOR method compared to the MSOR method requires less iteration steps and CPU times. This shows that when both RMSOR and MSOR methods are used to solve the LCP(q, A), the former is superior the latter. Table 2 presents the numerical results of the nonsymmetric case of Example 1. Specifically, some numerical results (including IT, CPU and RES) for the RMSOR method and the MSOR method for different problem sizes of m and the different values of µ are listed when r = 1.5 and t = 0.5. Table 2 shows that the RMSOR method and the MSOR method can still rapidly compute a satisfactory approximation to the solution of the LCP(q, A).  From Table 2, these numerical results further verify the observed results from Table 1. Our numerical experiments show that the computational efficiency of the RMSOR method is better than the MSOR method. It is noted that compared with the symmetric case, the iteration steps and the CPU times of the RMSOR method and the MSOR method slightly decrease in the nonsymmetric case. Table 3 presents the numerical results of Example 2. To compare the RMSOR method with the MSOR method, Table 3 lists the numerical results (including IT, CPU and RES) for the different values of µ under the same iteration parameter α. From the presented numerical results in Table 3, our numerical experiments show that that the RMSOR method requires less iteration steps and CPU times than the MSOR method. The numerical results in Table 3 confirm that the RMSOR method is still superior to the MSOR method.

Conclusions
In this paper, by introducing a relaxed matrix to obtain a new equivalent fixed-point form of the LCP(q, A), we establish a class of relaxed modulus-based matrix splitting (RMMS) iteration methods. Some sufficient conditions are presented to guarantee the convergence of RMMS iteration methods. Numerical examples show that the RMMS iteration method is feasible and overmatches the classical MMS iteration method under certain conditions. It is noted that our approach can be extended to other modulus-based matrix splitting methods, such as two-step modulus-based matrix splitting iteration methods, accelerated modulus-based matrix splitting methods, and so on.

Discussion
From our numerical experiments, we find that both the MMS iteration method and the RMMS iteration method are sensitive to the iteration parameter. This implies that the iteration parameter may play an important part in these two methods. Therefore, the determination of the optimal parameters for these two methods could be still an open problem, which is an interesting topic in the future.