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Journal of Applied Mathematics
Volume 2012 (2012), Article ID 743861, 14 pages
Generalized AOR Method for Solving Absolute Complementarity Problems
1Department of Mathematics, COMSATS Institute of Information Technology, Park Road, Chak Shahzad, Islamabad 44000, Pakistan
2Mathematics Department, College of Science, King Saud University, Riyadh 11451, Saudi Arabia
Received 25 January 2012; Accepted 9 February 2012
Academic Editor: Yonghong Yao
Copyright © 2012 Muhammad Aslam Noor et al. 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.
We introduce and consider a new class of complementarity problems, which is called the absolute value complementarity problem. We establish the equivalence between the absolute complementarity problems and the fixed point problem using the projection operator. This alternative equivalent formulation is used to discuss the existence of a solution of the absolute value complementarity problem. A generalized AOR method is suggested and analyzed for solving the absolute the complementarity problems. We discuss the convergence of generalized AOR method for the L-matrix. Several examples are given to illustrate the implementation and efficiency of the method. Results are very encouraging and may stimulate further research in this direction.
Complementarity theory introduced and studied by Lemke  and Cottle and Dantzig  has enjoyed a vigorous growth for the last fifty years. It is well known that both the linear and nonlinear programs can be characterized by a class of complementarity problems. The complementarity problems have been generalized and extended to study a wide class of problems, which arise in pure and applied sciences; see [1–24] and the references therein. Equally important is the variational inequality problem, which was introduced and studied in the early sixties. The theory of variational inequality has been developed not only to study the fundamental facts on the qualitative behavior of solutions but also to provide highly efficient new numerical methods for solving various nonlinear problems. For the recent applications, formulation, numerical results, and other aspects of the variational inequalities, see [13–22].
Motivated and inspired by the research going on in these areas, we introduced and consider a new class of complmenetarity problems, which is called the absolute value complmenetarity problem. Related to the absolute value complementarity problem, we consider the problem of solving the absolute value variational inequality. We show that if the under lying set is a convex cone, then both these problems are equivalent. If the underlying set is the whole space, then the absolute value problem is equivalent to solving the absolute value equations, which have been studied extensively in recent years.
We use the projection technique to show that the absolute value complementarity problems are equivalent to the fixed point problem. This alternative equivalent form is used to study the existence of a unique solution of the absolute value complementarity problems under some suitable conditions. We again use the fixed point formulation to suggest and analyze a generalized AOR method for solving the absolute value complementarity problems. The convergence analysis of the proposed method is considered under some suitable conditions. Some examples are given to illustrate the efficiency and implementation of the proposed iterative methods. Results are very encouraging. The ideas and the technique of this paper may stimulate further research in these areas.
Let be an inner product space, whose 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 and convex cone , , and will denote the vector in with absolute values of components of .
We remark that the absolute value complementarity problem (1.1) can be viewed as an extension of the complementarity problem considered by Lemke . To solve the linear complementarity problems, several methods were proposed. These methods can be divided into two categories, the direct and indirect (iterative) methods. Lemke  and Cottle and Dantzig  developed the direct methods for solving linear complementarity problems based on the process of pivoting, whereas Mangasarian , Noor [14, 15], and Noor et al. [20–22] considered the iterative methods. For recent applications, formulations, numerical methods, and other aspects of the complementarity problems and variational inequalities, see [1–24].
Let be a closed and convex set in the inner product space . We consider the problem of finding such that The problem (1.2) is called the absolute value variational inequality, which is a special form of the mildly nonlinear variational inequalities, introduced and studied by Noor  in 1975.
If , then the problem (1.2) is equivalent to find such that which are known as the absolute value equations. These equations have been considered and studied extensively in recent years; see [7–12, 17–20, 23, 24]. We would like to emphasize that Mangasarian [1, 2, 4–11] has shown that the absolute value equations (1.3) are equivalent to the complementarity problems (1.1). Mangasarian [7–11] has used the complementarity approach to solve the absolute value equations (1.3). For other methods, see [7–12, 20–24] and the references therein.
In this paper, we suggest a generalized AOR method for solving absolute complementarity problem, which is easy to implement and gives almost exact solution of (1.3).
We also need the following definitions and concepts.
Definition 1.1. is called an -matrix if for , and for .
Definition 1.2. A matrix is said to be positive definite matrix, if there exists a constant , such that and bounded if there exists a constant such that
2. Iterative Methods
To propose and analyze algorithm for absolute complementarity problems, we need the following well-known results.
Lemma 2.1 (see ). 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 .
Proof. Let be the solution of (1.2). Then
Since is a cone, taking and , we have
From inequality (1.2), we have
from which it follows that . Thus we conclude that is the solution of absolute complementarity problems (1.1).
Conversely, let be a solution of (1.1). Then From (2.5) and (2.6), it follows that Hence is the solution of absolute variational inequality (1.2).
In the next result, we prove the equivalence between the absolute value variational inequality (1.2) and the fixed point.
Lemma 2.3. Let be closed convex set in . Then, for a constant satisfies (1.2) if and only if satisfies the relation where is the projection of onto the closed convex set .
Now using Lemmas 2.2 and 2.3, we see that the absolute value complementarity problem (1.1) is equivalent to the fixed point problem of the following type: We use this alternative fixed point formulation to study the existence of a unique solution of the absolute value complementarity problem. Equation (1.1) and this is the main motivation of our next result.
Theorem 2.4. 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 .
Proof. Uniqueness: Let be two solutions of (1.2). Then
Taking in (2.13) and in (2.6), we have
Adding the previous inequalities, we obtain
which implies that
Since is positive definite, from (2.16), we have
As , therefore from (2.17) we have
which contradicts the fact that ; hence .
Let be the solution of (1.2). Then From Lemma 2.3, we have Define a mapping To show that the mapping defined by (2.21) has a fixed point, it is enough to prove that is a contraction mapping. For , consider where we have used the fact that is nonexpansive. Now using positive definiteness of , we have From (2.22) and (2.23), we have where Form and , we have .
This shows that is a contraction mapping and has a fixed point satisfying the absolute value variational inequality (1.2).
For the sake of simplicity, we consider the special case, when is a closed convex set in and we define the projection as We recall the following well-known result.
Lemma 2.5 (see ). For any and in , the projection has the following properties:(i), (ii), (iii),(iv).
We now suggest the iterative methods for solving the absolute value complmentarity problems (1.1). For this purpose, we decompose the matrix as, where is the diagonal matrix, and and are strictly lower and strictly upper triangular matrices, respectively. Let with and let be a real number.
Algorithm 2.6. Step 1. Choose an initial vector and a parameter ..Step 2. Calculate Step 3. If , then stop. Else, 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 value complementarity problem is nonempty.
To prove the convergence of Algorithm 2.6, we need the following result.
Theorem 2.7. Consider the operator as defined in (2.28). Assume that is an -matrix. Also assume that . Then, for any , the following holds:(i), (ii), (iii).
Proof. To prove (i), we need to verify that hold 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 (2.35) we can write . Hence (i) is proved.
Now we prove (ii). For this let us suppose that and . We will prove As so can be written as Similarly, for , we have and for , 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
We now consider the convergence criteria of Algorithm 2.6 and this is the main motivation of our next result.
Theorem 2.8. Assume that is an -matrix. Also assume that . Then for any initial vector , the sequence , defined by Algorithm 2.6 has the following properties:(i), (ii) is a unique solution of the absolute value complementarity problem (1.1).
Proof. Since , by (i) of Theorem 2.7, we have and . Recursively using Theorem 2.7 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 value complementarity problem (1.1).
3. Numerical Results
In this section, we consider several examples to show the efficiency of the proposed method. The convergence of the generalized AOR method is guaranteed for L-matrices only but it is also possible to solve different types of systems. The elements of the diagonal matrix are chosen from the interval such that where is the th diagonal element of . All the experiments are performed with Intel(R) Core(TM) 2 × 2.1 GHz, 1 GB RAM, and the codes are written in Matlab 7.
Example 3.1. We test Algorithm 2.6 on consecutively generated solvable random problems , and ranging from 10 to 1000. We chose a random matrix from a uniform distribution on , such that whose all diagonal elements are equal to 1000 and is chosen randomly from a uniform distribution on . The constant vector is computed as . The computational results are shown in Table 1.
Example 3.2. 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 -matrix. The comparison between the exact solution and the approximate solutions is given in Figure 1.
Example 3.3. 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 2.
In Table 2 TOC denotes total time taken by CPU. The rate of convergence of AOR method is better than iterative method .
In this paper, we have introduced a new class of complementarity problems, known as the absolute value complementarity problems. We have used the projection technique to establish the equivalence between the absolute value variational inequalities, fixed point problems, and the absolute value complementarity problems. This equivalence is used to discuss the existence of a unique solution of the absolute value problems under some suitable conditions. We have also used this alternative equivalent formulation to suggest and analyze an iterative method for solving the absolute value complementarity problems. Some special cases are also discussed. The results and ideas of this paper may be used to solve the variational inequalities and related optimization problems. This is another direction for future research.
This paper is supported by the Visiting Professor Program of King Saud University, Riyadh, Saudi Arabia. The authors are also grateful to Dr. S. M. Junaid Zaidi, Rector, COMSATS Institute of Information Technology, Pakistan, for providing the excellent research facilities.
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