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Journal of Applied Mathematics
Volume 2013, Article ID 172060, 6 pages
http://dx.doi.org/10.1155/2013/172060
Research Article

Symmetric SOR Method for Absolute Complementarity Problems

Department of Mathematics, Abdul Wali Khan University Mardan, Mardan 23200, Pakistan

Received 21 April 2013; Revised 3 August 2013; Accepted 19 August 2013

Academic Editor: Filomena Cianciaruso

Copyright © 2013 Javed Iqbal and Muhammad Arif. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We study symmetric successive overrelaxation (SSOR) method for absolute complementarity problems. Solving this problem is equivalent to solving the absolute value equations. Some examples are given to show the implementation and efficiency of the method.

1. Introduction

Absolute complementarity problem seeks real vectors and , such that where and . The complementarity theory was introduced and studied by Lemke [1] and Cottle and Dantzig [2]. The complementarity problems have been generalized and extended to study a wide class of problems, which arise in pure and applied sciences; see [19] and the references therein. Equally important is the variational inequality problem, which was introduced and studied in the early sixties.

In this paper, we suggest and analyze SSOR [5] method for absolute complementarity problem which was introduced by Noor et al. [10]. The convergence analysis of the proposed method is considered under some suitable conditions. We show that the absolute complementarity problems are equivalent to variational inequalities. Results are very encouraging. The ideas and the technique of this paper may stimulate further research in these areas.

Let be the finite dimension Euclidean space, whose the inner product and norm are denoted by and , respectively. For a given matrix , a vector , we consider the problem of finding , such that where is the polar cone of a closed convex cone in and will denote the vector in with absolute values of components of . We remark that the absolute value complementarity problem (2) can be viewed as an extension of the complementarity problem considered by Karamardian [6].

Let be a closed and convex set in the inner product space . We consider the problem of finding such that The problem (3) is called the absolute value variational inequality, which is a special form of the mildly nonlinear variational inequalities [11]. If , then the problem (3) is equivalent to find such that To propose and analyze algorithms for absolute complementarity problems, we need the following definitions.

Definition 1. is called an -matrix if for , and for , .

Definition 2. If is positive definite, then(i)there exists a constant , such that (ii)there exists a constant such that

2. Absolute Complementarity Problems

To propose and analyze algorithm for absolute complementarity problems, we need the following results.

Lemma 3 (see [12]). Let be a nonempty closed convex set in . For a given satisfies the inequality if and only if where is the projection of onto the closed convex set .

Lemma 4 (see [10]). If is the positive cone in , then is a solution of absolute variational inequality (3) if and only if is the solution of the complementarity problem (2).

The next result proves the equivalence between variational inequality (3) and the fixed point.

Lemma 5 (see [10]). If is closed convex set in , then ; satisfies (3) if and only if satisfies the relation where is the projection of onto the closed convex set .

Now using Lemmas 4 and 5, the absolute complementarity problem (2) can be transformed to fixed-point problem as

Theorem 6 (see [10]). Let be a positive definite matrix with constant and continuous with constant . If , , , then there exists a unique solution , such that where is a closed convex set in .

To define the projection operator , we consider the special case when is a closed convex set in , as follows.

Definition 7 (see [3]). Let is a closed convex set in . Then, the projection operator is defined as

Lemma 8 (see [3]). For any and in , the following facts hold:(i);(ii);(iii);(iv), with equality, if and only if .
Now one splits the matrix as where is the diagonal matrix and and are strictly lower and strictly upper triangular matrices, respectively. Let ; using (13), one suggests the SSOR method for solving (3) as follows.

Algorithm 9. Consider the following.

Step  1. Choose an initial vector and a parameter . Set .

Step  2. Calculate

Step  3. If , then stop; else, set and go to step  2.

Algorithm 10. Consider the following.

Step  1. Choose an initial vector and a parameter . Set .

Step  2. Calculate

Step  3. If , then stop; else, set and go to step  2.

Now we define an operator such that , where is the fixed point of the system We also assume that the set of the absolute complementarity problem is nonempty. To prove the convergence of Algorithm 9, we need the following result.

Theorem 11. Consider the operator as defined in (16). Assume that is an -matrix. Also assume that . Then for any , it holds that.(i);(ii);(iii).

Proof. To prove , we need to prove that with satisfying To prove the required result, we use mathematical induction. For this, let :
Since , ; therefore, .
For , we have Here, , , and . This implies that .
Suppose that we have to prove that the statement is true for ; that is, Consider Since , , and for ; from (24), we can write Hence, is proved.
Now we prove , for this let us suppose that and . We will prove that
As so can be written as Similarly, for we have For , we have Since , therefore . Hence, it is true for . Suppose it is true for ; we will prove it for ; for this consider Since and for , hence it is true for and (ii) is verified.
Next we prove (iii); that is,
Let from (i) . Also by definition of and .
Now For by definition of . Suppose that , so which contradicts the fact that . Therefore, .
Now we prove it for any in . Suppose the contrary ; then As it is true for all , it should be true for . That is, which contradicts the fact that . So, , for any in .
Hence, .

Now we prove the convergence criteria of Algorithm 9 when the matrix is an -matrix as stated in the next result.

Theorem 12. Assume that is an -matrix and . Then for any initial vector , the sequence , defined by Algorithm 9 has the following properties:(i); ;(ii) is the unique solution of the absolute complementarity problem.

Proof. Since , by (i) of Theorem 11, we have and . Recursively using Theorem 11, we obtain From (i), we observe that the sequence is monotone bounded; therefore, it converges to some satisfying Hence, is the solution of the absolute complementarity problem (2).

Note. The convergence of Algorithm 10 has the same steps as given in Theorems 11 and 12.

3. Numerical Results

In this section, we consider several examples to show the efficiency of the proposed methods. The convergence of SSOR method is guaranteed for -matrices only, but it is also possible to solve different type of systems. All the experiments are performed with Intel(R) Core 2 × 2.1 GHz, 1 GB RAM, and the codes are written in MATLAB 7.

Example 13 (see [10]). Consider the ordinary differential equation The exact solution is
We take ; the matrix is given by
The constant vector is given by Here, is not an -matrix. The comparison between the exact solution and the approximate solutions is given in Figure 1.
In Figure 1, we see that the SSOR method converges rapidly to the approximate solution of absolute complementarity problem (2) as compared to GAOR method.
In the next example, we compare SSOR method with iterative method by Noor et al. [13].

172060.fig.001
Figure 1: Comparison between GAOR method and SSOR method.

Example 14 (see [13]). Let the matrix be given by Let , the problem size , ranging from 4 to 1024. The stopping criteria are . We choose initial guess as . The computational results are shown in Table 1.
In Table 1, TOC denotes the total time taken by CPU. The rate of convergence of SSOR method is better than that of iterative method [13].

tab1
Table 1

4. Conclusion

In this paper, we have discussed symmetric SOR method for solving absolute complementarity problem. The comparison with other methods showed the efficiency of the method. The results and ideas of this paper may be used to solve the variational inequalities and related optimization problems.

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