GLOBAL RELAXED MODULUS-BASED SYNCHRONOUS BLOCK MULTISPLITTING MULTI-PARAMETERS METHODS FOR LINEAR COMPLEMENTARITY PROBLEMS

Recently, Bai and Zhang [Numerical Linear Algebra with Applications, 2013, 20, 425–439] constructed modulus-based synchronous multisplitting methods by an equivalent reformulation of the linear complementarity problem into a system of fixed-point equations and studied the convergence of them; Li et al. [Journal of Nanchang University (Natural Science), 2013, 37, 307–312] studied synchronous block multisplitting iteration methods; Zhang and Li [Computers and Mathematics with Application, 2014, 67, 1954–1959] analyzed and obtained the weaker convergence results for linear complementarity problems. In this paper, we generalize their algorithms and further study global relaxed modulus-based synchronous block multisplitting multiparameters methods for linear complementarity problems. Furthermore, we give the weaker convergence results of our new method in this paper when the system matrix is a block H+−matrix. Therefore, new results provide a guarantee for the optimal relaxation parameters, please refer to [A. Hadjidimos, M. Lapidakis and M. Tzoumas, SIAM Journal on Matrix Analysis and Applications, 2012, 33, 97–110, (dx.doi.org/10.1137/100811222)], where optimal parameters are determined.


Introduction
Consider the linear complementarity problem, abbreviated as LCP(q, A), for finding a pair of real vectors r and z ∈ R n such that r := Az + q ≥ 0, z ≥ 0 and z T (Az + q) = 0, (1.1) where A = (a ij ) ∈ R n×n is a given large, sparse and real matrix and q = (q 1 , q 2 , ..., q n ) T ∈ R n is a given real vector. Here, z T and ≥ denote the transpose of the vector z and the componentwise defined partial ordering between two vectors, respectively. Many problems in scientific computing and engineering applications may lead to solutions of LCPs of the form (1.1). For example, the linear complementarity problem may arise from application problems such as the convex quadratic programming, the Nash equilibrium point of the bimatrix game, the free boundary problems of fluid dynamics etc. (e.g. see [15,17] and the references therein). Some solvers for LCP(q, A) with a special matrix A were proposed [2-8, 14, 16, 20]. Recently, many people have focused the solver of LCP(q, A) with an algebra equation [7-9, 11-14, 16, 20, 29, 33-42]. In particular Bai proposed a modulus-based matrix multisplitting iteration method for solving LCP(q, A) and presented convergence analysis for the proposed methods; see [7,8]. Zhang and Ren [33] extended the condition of a compatible H−splitting to that of an H-splitting. Li [27] extended the modulus-based matrix splitting iteration method to more general cases. Bai [10] presented parallel matrix block multisplitting relaxation iteration methods and established the convergence theory of these new methods in a thorough manner. Li et al. [28] studied synchronous block multisplitting iteration methods. Zhang and Li [35] generalized Bai and Zhang's methods [1] and studied modulus-based synchronous multisplitting multi-parameters methods for linear complementarity problems.
In this paper, we generalize the methods of Bai and Zhang's [1] and Zhang and Li's [35] from point form to block form according to the modulus-based synchronous multisplitting iteration methods and consider global relaxed modulus-based synchronous block multisplitting multi-parameters method for solving LCP(q, A). Moreover, we give some theoretical analysis and improve some existing convergence results in [1,28].
The rest of this paper is organized as follows: In section 2, we give some notations and lemmas. In section 3, we propose global relaxed modulus-based synchronous block multisplitting multi-parameters method for solving LCP(q, A). In section 4, we give the convergence analysis for the proposed method.

Lemma 2.11 ( [4]).
A ∈ R n×n be an H + -matrix. Then, the LCP(q, A) has a unique solution for any q ∈ R n . Lemma 2.12 ( [7]). Let A = M − N be a splitting of the matrix A ∈ R n×n , Ω be a positive diagonal matrix, and γ a positive constant. Then, for the LCP(q, A) the following statements hold true: is a solution of the LCP(q, A).

GRMSBMMAOR methods
At first, we introduce the concept of multisplitting method and the detailed process of parallel iterative method.
Given a positive diagonal matrix Ω and a positive constant γ, form Lemma 2.13, we know that if x satisfies either of the implicit fixed-point equations is a solution of the LCP(q, A). Based on block matrix A ∈ R m×m , the corresponding block diagonal matrix is D = diag(A 11 , A 22 , ..., A pp ), and L k is block strictly triangular matrix, With the equivalent reformulations (4), (5) and accelerated over-relaxation (AOR) of the LCP(q, A), we can establish the following global relaxed modulus-based synchronous block multisplitting multi-parameters AOR method (GRMSBMMAOR), which is similar to Method 3.1 in [19] and Method 3.1 in [28].

Convergence analysis
In 2013, based on the modulus-based synchronous multisplitting AOR method, Bai and Zhang [1] obtained the following results.  1, 2, ..., l) be a multisplitting and a triangular multisplitting of the matrix A, respectively. Assume that γ > 0 and the positive diagonal matrix Ω satisfies Ω ≥ D. 1, 2, ..., l), then the iteration sequence {z (m) } ∞ m=0 generated by the MSMAOR iteration method converges to the unique solution z * of LCP(q, A) for any initial vector z (0) ∈ R n + , provided the relaxation parameters α and β satisfy In 2013, based on the modulus-based synchronous block multisplitting AOR method, Li et al. [28] analyzed the following results. .
In 2014, based on the modulus-based synchronous multisplitting multi-parameters AOR method, Zhang and Li [35] studied the following results.
Based global relaxed modulus-based synchronous block multisplitting multiparameters AOR method, we will present a weaker convergence results of the multisplitting methods for the linear complementarity problem when the system matrix is a block H + -matrix, which is as follows:  1, 2, ..., l), then the iteration sequence {z (m) } ∞ m=0 generated by the GRMSBMMAOR iteration method converges to the unique solution z * of LCP(q, A) for any initial vector z (0) ∈ R n + , provided the relaxation parameters α k and β k satisfy Proof. From Lemma 2.11 and (3.3), for the GRMSBMMAOR method, it holds that By subtracting (4.2) from (3.3), we have (4.5) The error relationship (4.4) is the base for proving the convergence of GRMSB-MMAOR method. By making use of Lemmas 2.5 and 2.6, defining ϵ (m) = x (m) − x * and arranging similar terms together, we can obtain (4.6) (4.7) So, we have . Through further analysis, we have Since A ∈ L n,I (n 1 , n 2 , ..., n p ) is a block H  = (1, 1, ..., 1) T ∈ R n . Since J ϵ is nonnegative, the matrix J ⟨H⟩ + ϵee T has only positive entries and irreducible for any ϵ > 0. By the Perron-Frobenius theorem for any ϵ > 0, there is a vector x ϵ > 0 such that where ρ ϵ = ρ(J ⟨H⟩ + ϵee T ) = ρ(J ϵ ). Moreover, if ϵ > 0 is small enough, we have ρ ϵ < 1 by continuity of the spectral radius. Because of 0 < α k ≤ 1, we also have 1 − 2α k + 2α k ρ < 1, and 1 − 2α k + 2α k ρ ϵ < 1. So Multiplying x ϵ in two sides of the above inequality, and M −1 k ≥ D −1 ⟨H⟩ , we can obtain Based on E k and the definition of [•], we know that Similar to the Case 1, let e denote the vector e = (1, 1, ..., 1) T ∈ R n , and x ϵ > 0 such that J ϵ x ϵ = (J ⟨H⟩ + ϵee T )x ϵ = ρ(J ϵ )x ϵ . Moreover, if ϵ > 0 is small enough, we have ρ ϵ < 1 by continuity of the spectral radius. Because of 1 < α k < 1 µ1(P AQ) , we also have Multiplying x ϵ in two sides of the above inequality, and M −1 Based on E k and the definition of [•], we know that l ∑ k=1 [E k ] = I. By (4.5), we have Similar to the Case 1, let e denote the vector e = (1, 1, ..., 1) T ∈ R n , and Moreover, if ϵ > 0 is small enough, we have ρ ϵ < 1 by continuity of the spectral radius. Because of 0 < β k < 1 µ1(P AQ) , we also have 2β k ρ − 1 < 1 and 2β k ρ ϵ − 1 < 1. So . Multiplying x ϵ in two sides of the above inequality, and M −1 k ≥ D −1 ⟨H⟩ , we can obtain Based on E k and the definition of [•], we know that where θ 3 = ωρ ′ + |1 − ω| < 1.

Remark 4.1.
Obviously, from Figure 1, we can find that the conditions of Theorem 4.4 (when ω = 1) in this paper are weaker than those of Theorem 1 in [23]. Moreover, the parameters can be adjusted suitably so that the convergence property of method can be substantially improved. That is to say, we have more choices for the splitting A = B − C which makes the multisplitting iteration methods converge. Therefore, our convergence theories extend the scope of multisplitting iteration methods in applications. Based on the similar proving process of Theorem 4.4, we can obtain the following convergence results.  Table 1, we obviously see that the MSMMAOR method in [1] and the MSBMAOR method in [28] use the same parameters α, β in different processors, but the GRMSBMMAOR method in this paper uses different parameters α k , β k (k = 1, 2, ..., l) in different processors. Moreover, when computing x (m+1) in Method 3.1, we utilize relaxation extrapolation technique and add a relaxation parameter ω. Therefore, we may choose proper relaxation parameters to increase convergence speed and reduce the computation time when doing numerical experiments.