Generalized AOR Method for Solving Absolute Complementarity Problems

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.


Introduction
Complementarity theory introduced and studied by Lemke 1 and Cottle and Dantzig 2 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 R n be an inner product space, whose inner product and norm are denoted by •, • and • , respectively.For a given matrix A ∈ R n×n , a vector b ∈ R n , we consider the problem of finding where K * {x ∈ R n : x, y ≥ 0, ∀y ∈ K} is the polar cone of a closed and convex cone K, A ∈ R n×n , b ∈ R n , and |x| will denote the vector in R n with absolute values of components of x ∈ R n .We remark that the absolute value complementarity problem 1.1 can be viewed as an extension of the complementarity problem considered by Lemke 1 .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 1 and Cottle and Dantzig 2 developed the direct methods for solving linear complementarity problems based on the process of pivoting, whereas Mangasarian 10 ,Noor 14,15 , considered the iterative methods.For recent applications, formulations, numerical methods, and other aspects of the complementarity problems and variational inequalities, see 1-24 .
Let K be a closed and convex set in the inner product space R n .We consider the problem of finding x ∈ K 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 13  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.

Iterative Methods
To propose and analyze algorithm for absolute complementarity problems, we need the following well-known results.
if and only if Since K is a cone, taking y 0 ∈ K and y 2x ∈ K, we have From inequality 1.2 , we have from which it follows that Ax − |x| − b ∈ K * .Thus we conclude that x ∈ K is the solution of absolute complementarity problems 1.1 .
Conversely, let x ∈ K be a solution of 1.1 .Then From 2.5 and 2.6 , it follows that Hence x ∈ K is the solution of absolute variational inequality 1.2 .
From Lemma 2.1, it follows that both the problems 1.1 and 1.2 are equivalent.
In the next result, we prove the equivalence between the absolute value variational inequality 1.2 and the fixed point.
where P K is the projection of R n onto the closed convex set K.
Proof.Let x ∈ K be the solution of 1.2 .Then, for a positive constant ρ > 0, Using Lemma 2.1, we have which is the required result.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:

2.11
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 A ∈ R n×n be a positive definite matrix with constant α > 0 and continuous with constant where K is a closed convex set in R n .
Proof.Uniqueness: Let x 1 / x 2 ∈ K be two solutions of 1.2 .Then Taking y x 2 ∈ K in 2.13 and y x 1 ∈ K in 2.6 , we have Adding the previous inequalities, we obtain Since A is positive definite, from 2.16 , we have As γ > 1, therefore from 2.17 we have which contradicts the fact that

2.19
From Lemma 2.3, we have Define a mapping To show that the mapping F x defined by 2.21 has a fixed point, it is enough to prove that F x is a contraction mapping.For

2.22
where we have used the fact that P K is nonexpansive.Now using positive definiteness of A, we have
For the sake of simplicity, we consider the special case, when K 0, c is a closed convex set in R n and we define the projection P K x as

2.25
We recall the following well-known result.
Lemma 2.5 see 3 .For any x and y in R n , the projection P K x has the following properties: We now suggest the iterative methods for solving the absolute value complmentarity problems 1.1 .For this purpose, we decompose the matrix A as, where D is the diagonal matrix, and L and U are strictly lower and strictly upper triangular matrices, respectively.Let Ω diag ω 1 , ω 2 , . . ., ω n with ω i ∈ R and let α be a real number.
Step 1. Choose an initial vector x 0 ∈ R n and a parameter Ω ∈ R .Set k 0.

2.27
Step 3. If x k 1 x k , then stop.Else, set k k 1 and go to Step 2. Now we define an operator g : R n → R n such that g x ξ, where ξ is the fixed point of the system

2.28
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 g : R n → R n as defined in 2.28 .Assume thatA ∈ R n×n is an L-matrix.Also assume that 0 < ω i ≤ 1, 0 ≤ α ≤ 1.Then, for any x ∈ ϕ, the following holds: Proof.To prove i , we need to verify that ξ i ≤ x i , i 1, 2, . . ., n hold with ξ i satisfying

2.30
To prove the required result, we use mathematical induction.For this let i 1,

2.33
We have to prove that the statement is true, for i k, that is,

2.36
As so ξ i can be written as

2.38
Similarly, for φ i , we have

2.39
and for i 1,

2.40
Since Hence it is true for i 1. Suppose it is true for i 1, 2, . . .k − 1; we will prove it for i k; for this consider

2.41
Since x ≤ y, and ξ i ≤ φ i for i 1, 2, . . .k − 1, hence it is true for k and ii is verified.
Next we prove iii , that is, Also by definition of g, ξ g x ≥ 0 and λ g ξ ≥ 0. Now

2.45
As it is true for all α ∈ 0, 1 , it should be true for α 0. That is, 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 A ∈ R n×n is an L-matrix.Also assume that 0 < ω i ≤ 1, 0 ≤ α ≤ 1.Then for any initial vector x 0 ∈ ϕ, the sequence {x k }, k 0, 1, 2, . .., defined by Algorithm 2.6 has the following properties:  Proof.Since x 0 ∈ ϕ, by i of Theorem 2.7, we have x 1 ≤ x 0 and x 1 ∈ ϕ.Recursively using Theorem 2.7 we obtain 0 ≤ x k 1 ≤ x k ≤ x 0 ; k 0, 1, 2, . . . .

2.47
From i we observe that the sequence {x k } is monotone bounded; therefore, it converges to some x * ∈ R n satisfying

2.48
Hence x * is the solution of the absolute value complementarity problem 1.1 .

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 Ω are chosen from the interval a, b such that where ω i is the ith diagonal element of Ω.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 m consecutively generated solvable random problems A ∈ R n×n , and n ranging from 10 to 1000.We chose a random matrix A from a uniform distribution on 0, 1 , such that whose all diagonal elements are equal to 1000 and x is chosen randomly from a uniform distribution on 0, 1 .The constant vector is computed as b Ax − |x|.The computational results are shown in Table 1.Example 3.2.Consider the ordinary differential equation: The exact solution is We take n 10; the matrix A is given by

3.4
The

3.5
Here A is not L-matrix.The comparison between the exact solution and the approximate solutions is given in Figure 1.
In Table 2 TOC denotes total time taken by CPU.The rate of convergence of AOR method is better than iterative method 21 .

Conclusion
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.
is a unique solution of the absolute value complementarity problem 1.1 .