A TWO-STEP MODULUS-BASED MULTISPLITTING ITERATION METHOD FOR THE NONLINEAR COMPLEMENTARITY PROBLEM∗

In this paper, we construct a two-step modulus-based multisplitting iteration method based on multiple splittings of the system matrix for the nonlinear complementarity problem. And we prove its convergence when the system matrix is an H-matrix with positive diagonal elements. Numerical experiments show that the proposed method is efficient.


Introduction
For a given matrix A ∈ R n×n and vector q ∈ R n , the nonlinear complementarity problem N CP (A, q) consists of finding a vector z ∈ R n which satisfies the conditions z ≥ 0, Az + q + φ(z) ≥ 0, z T (Az + q + φ(z)) = 0.
(1.1) papers [5] and [10], the authors presented accelerated modulus-based matrix splitting iteration methods to solve a class of nonlinear complementarity problems. Ke, Ma and Zhang [8] established a class of relaxation modulus-based matrix splitting iteration methods for circular cone nonlinear complementarity problems. In addition, Ke and Ma [6] analyzed the convergence of the two-step modulusbased matrix splitting iteration method for LCP and they presented the convergence conditions. Bai and Zhang [2] constructed modulus-based multisplitting iteration methods for LCP based on multiple splittings of the system matrix and they presented the convergence theory. Li, Wang and Yin [9] gave the two-step modulusbased matrix splitting iteration method for a restricted class of NCP. In this paper, we construct a two-step modulus-based multisplitting iteration method based on multiple splittings of the system matrix for NCP. This paper is organized as follows. Section 2 is the preliminaries. In Section 3, the two-step modulus-based multisplitting iteration method for NCP is introduced. The convergence of this method for H-matrices is considered in Section 4. One numerical example is given in Section 5.

Preliminaries
For convenience, we first briefly describe the notations.
Let A ∈ R n×n be an n × n matrix, for A,
To suit computational requirements of the modern high-speed multiprocessor systems, by Lemma 3.1, we establish the following two-step modulus-based multisplitting(TMM) iteration method and its several special explicit forms.
Step 4. If z (m+1) satisfies a prescribed stopping rule, then stop. Otherwise, set m := m + 1 and return to Step 2.

Main Results
To present the following discussion, we assume that By the differential mean value theorem, there exists ξ and Ω is a positive diagonal matrix satisfying Ω ≥ D + ψ, then for any initial vector x (0) ∈ R n the iterative sequence z (m) ∞ m=0 generated by the TMM method convergences to the unique solution z * of the N CP (A, q).
Proof. Let z * be a solution of (1.1), then To prove lim m→∞ z (m) = z * , we need only to prove that lim m→∞ x (m) = x * . By (3.2) and (4.1), we have So the error formula of the TMM iteration method is Since l TMM is nonnegative, by Lemma 2.3, we have ρ (l TMM ) < 1.
The proof is completed. From Theorem 4.1, we can obtain the following theorem easily.

Theorem 4.2. Let A ∈ R n×n be an H-matrix with positive diagonal elements,
Assume that ψ k ≤ ψ, h > 0 and Ω is a positive diagonal matrix satisfying Ω ≥ D + ψ, then for any initial vector x (0) ∈ R n , the iterative sequence z (m) ∞ m=0 generated by the TMMAOR method convergences to the unique solution z * of the N CP (A, q), provided that 0 < β ≤ α ≤ 1.

Numerical example
One numerical example is given in this section to illustrate the efficiency of the proposed method and to verify the convergence theory established above. In all the following numerical experiments, the initial vector is chosen to be zero and h = 1. And set A = D − L − U , where D, −L, −U are the diagonal, the strictly lower-triangular and the strictly upper-triangular matrices of A, respectively. Let Since the complementarity condition z T (Az + q + φ(z)) = 0 is equivalent to min Az (k) + q + φ z (k) , z (k) 2 = 0, iterations are terminated when the norm of the residual vector (denoted by 'RES') satisfies RES ≤ 10 −5 , or k reaches the maximal number of iteration steps, which is 1000 in our paper. All the computations are performed in MATLAB ® with double machine precision where the CPU is 2.40 GHz and the memory is 4.00 GB.
Example 5.1 ( [9]). Let m be a given positive integer, n = m 2 . Choose A in (1.1) to be a block upper tridiagonal matrix as follows: where S = tridiag(−1, 4, −1) ∈ R m×m is a tridiagonal matrix. Let q = (1, −1, · · · , 1, (−1) n−1 ) T ∈ R n and φ(z) = z 2 1 + 0.25, z 2 2 + 0.25, · · · , z 2 n + 0.25 The matrix A in Example 5.1 is an H + -matrix. In actual implementation, the parameter matrix Ω is chosen to be D + I in Example 5.1 for both the two-step modulus-based multisplitting successive overrelaxation method and the two-step modulus-based successive overrelaxation method, where D is the diagonal matrix of A, I is the identity matrix. For TMMSOR, we choose E 1 = diag(1, 0, 1, 0, · · · , n mod 2) ∈ R n×n and E 2 = I − E 1 .  Table 1, the number of iteration steps (denoted by 'IT') and the elapsed CPU time in seconds (denoted by 'CPU') are listed for the two-step modulus-based multisplitting successive overrelaxation iteration method and the two-step modulusbased successive overrelaxation iteration method when parameter α varies from 0.8 to 1.4 with m = 256. The optimal parameters α * is chosen firstly to minimize the  Table 1, it is seen that for Example 5.1, the optimal parameter α * = 1.1 for both the two-step modulus-based multisplitting successive overrelaxation iteration method and the two-step modulus-based successive overrelaxation iteration method when m = 256. In the following, we choose α * = 1.1 for both the two-step modulusbased multisplitting successive overrelaxation iteration method and the two-step modulus-based successive overrelaxation iteration method.
In Table 2, the number of iteration steps, the elapsed CPU time in seconds and the residual for four methods are listed respectively when m is varying.
From Table 2, it is observed that with the same dimension, the number of iteration steps for two-step modulus-based multisplitting method is less than that for modulus-based matrix splitting method and two-step modulus-based matrix splitting method, and the two-step modulus-based multisplitting method costs less CPU time. Meanwhile, the CPU time increases when the problem size n = m 2 increases for all methods, while the number of the iteration steps changes few.

Conclusions
In this paper, the two-step modulus-based multisplitting iteration method for a class of nonlinear complementarity problems was proposed and its convergence theories were studied when the system matrix is an H-matrix with positive diagonal elements. Numerical experiments showed the new method is more effective than modulus-based matrix splitting method and two-step modulus-based matrix splitting method.