# Residual Iterative Method for Solving Absolute Value Equations

Authors: Muhammad Aslam Noor; Javed Iqbal; Eisa Al-Said
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

We suggest and analyze a residual iterative method for solving absolute value equationsAx-x=b where A∈Rn×n, b∈Rn are given and x∈Rn is unknown, using the projection technique. We also discuss the convergence of the proposed method. Several examples are given to illustrate the implementation and efficiency of the method. Comparison with other methods is also given. Results proved in this paper may stimulate further research in this fascinating field.


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## Body

## 1. Introduction

The residual methods were proposed for solving large sparse systems of linear equations(1.1)Ax=b,
where A∈Rn×n is a positive definite matrix and x,b∈Rn. Paige and Saunders [1] minimized the residual norm over the Krylov subspace and proposed an algorithm for solving indefinite systems. Saad and Schultz [2] used Arnoldi process and suggested generalized minimal residual method which minimize norm of the residual at each step. The residual methods have been studied extensively [3–5].We show that the Petrov-Galerkin process can be extended for solving absolute value equations of the form:(1.2)Ax-|x|=b,
where A∈Rn×n, b∈Rn. Here |x| is the vector in Rn with absolute values of the components of x and x∈Rn, is unknown. The absolute value equations (1.1) were investigated extensively in [6]. It was Managasarian [7, 8], who proved that the absolute value equations (1.2) are equivalent to the linear complementarity problems. This equivalent formulation was used by Managasarian [7, 8] to solve the absolute value equations. We would like to remark that the complementarity problems are also equivalent to the variational inequalities. Thus, we conclude that the absolute value equations are equivalent to the variational inequalities. There are several methods for solving the variational inequalities; see Noor [9–11], Noor et al. [12, 13] and the references therein. To the best our knowledge, this alternative equivalent formulation has not exploited up to now. This is another direction for future direction. We hope that this interlink among these fields may lead to discover novel and innovative techniques for solving the absolute value equations and related optimization problems. Noor et al. [14, 15] have suggested some iterative methods for solving absolute value equation (1.2) using minimization technique with symmetric positive definite matrix. For more details, see [3, 4, 6–12, 14–19].In this paper, we suggest and analyse residual iterative method for solving absolute value equations (1.2) using projection technique. Our method is easy to implement. We discuss the convergence of the residual method for nonsymmetric positive definite matrices.We denote byK and L the search subspace and the constraints subspace, respectively, and let m be their dimension and x0∈Rn an initial guess. A projection method onto the subspace K and orthogonal to L is a process to find an approximate solution x∈Rn to (1.2) by imposing the Petrov-Galerkin conditions that x belong to affine space x0+K such that the new residual vector orthogonal to L, that is,
(1.3)findx∈x0+Ksuchthatb-(A-D(x))x⊥L,
where D(x) is diagonal matrix corresponding to sign(x). For different choices of the subspace L, we have different iterative methods. Here we use the constraint space L=(A-D(x))K. The residual method approximates the solution of (1.2) by the vector x∈x0+Kthat minimizes the norm of residual.The inner product is denoted by〈·,·〉 in the n-dimensional Euclidean space Rn. For x∈Rn, sign(x) will denote a vector with components equal to 1,0,-1 depending on whether the corresponding component of x is positive, zero, or negative. The diagonal matrix D(x) corresponding to sign(x) is defined as(1.4)D(x)=∂|x|=diag⁡(sign⁡(x)),
where ∂|x| represent the generalized Jacobean of |x| based on a subgradient [20, 21].We denote the following by(1.5)a=〈Cv1,Cv1〉,c=〈Cv1,Cv2〉,d=〈Cv2,Cv2〉,p1=〈b-Axk+|xk|,Cv1〉=〈b-Cxk,Cv1〉,p2=〈b-Axk+|xk|,Cv2〉=〈b-Cxk,Cv2〉,
where 0≠v1, v2∈Rn, and C=A-D(xk). We consider A such that C is a positive definite matrix. We remark that D(xk)xk=|xk|.


## 2. Residual Iterative Method

Consider the iterative scheme of the type:(2.1)xk+1=xk+αv1+βv2,0≠v1,v2∈Rn,k=0,1,2,….
These vectors can be chosen by different ways. To derive residual method for solving absolute value equations in the first step, we choose the subspace(2.2)K1=span⁡{v1},L1=span⁡{Cv1},x0=xk.
For D(x̃k+1)=D(xk), we write the residual in the following form:
(2.3)b-Ax̃k+1+|x̃k+1|=b-(A-D(x̃k+1))x̃k+1=b-(A-D(xk))x̃k+1=b-Cx̃k+1.
From (1.3) and (2.3), we calculate
(2.4)x̃k+1∈xk+K1suchthatb-Cx̃k+1⊥L1;
that is, we find the approximate solution by the iterative scheme
(2.5)x̃k+1=xk+αv1.
Now, we rewrite (2.4) in the inner product as
(2.6)〈b-Cx̃k+1,Cv1〉=0;
from the above discussion, we have
(2.7)〈b-Cxk-αCv1,Cv1〉=〈b-Cxk,Cv1〉-α〈Cv1,Cv1〉=p1-aα=0,
from which we have
(2.8)α=p1a.
The next step is to choose the subspace
(2.9)K2=span⁡{v2},L2=span⁡{Cv2},x0=x̃k+1,
and to find the approximate solution xk+1 such that
(2.10)xk+1∈x̃k+1+K2suchthatb-Cxk+1⊥L2,
where
(2.11)xk+1=x̃k+1+βv2,b-Axk+1+|xk+1|=b-Cxk+1,D(xk+1)=D(xk).
Rewriting (2.10) in terms of the inner product, we have(2.12)〈b-Cxk+1,Cv2〉=0.
Thus, we have(2.13)〈b-Cxk+1,Cv2〉=〈b-Cxk-αCv1-βCv2,Cv2〉=〈b-Cxk,Cv2〉-α〈Cv1,Cv2〉-β〈Cv2,Cv2〉=p2-cα-dβ=0.
From (2.8) and (2.13), we obtain(2.14)β=ap2-cp1ad.We remark that one can choosev1=rk and v2 in different ways. However, we consider the case v2=sk (sk is given in Algorithm 2.1).Based upon the above discussion, we suggest and analyze the following iterative method for solving the absolute value equations (1.2) and this is the main motivation of this paper.Algorithm 2.1.
Choose an initial guessx0∈Rn,Fork=0,1,2,… until convergence dork=b-Axk+|xk|gk=(A-D(xk))T(Axk-|xk|-b)Hk=((A-D(xk))-1(A-D(xk)))Tsk=-HkgkIf∥rk∥=0, then stop; elseαk=p1/a,βk=(ap2-cp1)/adSetxk+1=xk+αkrk+βkskIf∥xk+1-xk∥<10-6then stopEnd ifEnd fork.Ifβ=0, then Algorithm 2.1 reduces to minimal residual method; see [2, 5, 21, 22]. For the convergence analysis of Algorithm 2.1, we need the following result.Theorem 2.2.
Let{xk} and {rk} be generated by Algorithm 2.1; if D(xk+1)=D(xk), then
(2.15)‖rk‖2-‖rk+1‖2=p12a+(ap2-cp1)2a2d,
where rk+1=b-Axk+1+|xk+1| and D(xk+1)=diag(sign(xk+1)).Proof.
Using (2.1), we obtain
(2.16)rk+1=b-Axk+1+|xk+1|=b-(A-D(xk+1))xk+1=b-(A-D(xk))xk+1=b-(A-D(xk))xk-α(A-D(xk))v1-β(A-D(xk))v2=b-Axk+|xk|-αCv1-βCv2=rk-αCv1-βCv2.
Now consider
(2.17)‖rk+1‖2=〈rk+1,rk+1〉=〈rk-αCv1-βCv2,rk-αCv1-βCv2〉=〈rk,rk〉-2α〈rk,Cv1〉-2αβ〈Cv1,Cv2〉-2β〈rk,Cv2〉+α2〈Cv1,Cv1〉+β2〈Cv2,Cv2〉=‖rk‖2-2αp1+2cαβ-2βp2+aα2+β2d.
From (2.8), (2.14), and (2.17), we have
(2.18)‖rk‖2-‖rk+1‖2=p12a+(ap2-cp1)2a2d,
the required result (2.15).Sincep12/a+(ap2-cp1)2/a2d≥0, so from (2.18) we have
(2.19)‖rk‖2-‖rk+1‖2=p12a+(ap2-cp1)2a2d≥0.
From (2.19) we have ∥rk+1∥2≤∥rk∥2. For any arbitrary vectors 0≠v1,v2∈Rn, α,β are defined by (2.8), and (2.14) minimizes norm of the residual.We now consider the convergence criteria of Algorithm2.1, and it is the motivation of our next result.Theorem 2.3.
IfC is a positive definite matrix, then the approximate solution obtained from Algorithm 2.1 converges to the exact solution of the absolute value equations (1.2).Proof.
From (2.15), we have
(2.20)‖rk‖2-‖rk+1‖2≥p12a=〈rk,Crk〉2〈Crk,Crk〉≥λmin⁡2‖rk‖4λmax⁡2‖rk‖2=λmin⁡2λmax⁡2‖rk‖2.
This means that the sequence ∥rk∥2 is decreasing and bounded. Thus the above sequence is convergent which implies that the left-hand side tends to zero. Hence ∥rk∥2 tends to zero, and the proof is complete.


## 3. Numerical Results

To illustrate the implementation and efficiency of the proposed method, we consider the following examples. All the experiments are performed with Intel(R) Core(TM) 2 × 2.1 GHz, 1 GB RAM, and the codes are written in Mat lab 7.Example 3.1.
Consider the ordinary differential equation:(3.1)d2xdt2-|x|=(1-t2),0≤t≤1,x(0)=-1x(1)=0.
We discredited the above equation using finite difference method to obtain the system of absolute value equations of the type:
(3.2)Ax-|x|=b,
where the system matrix A of size n=10 is given by
(3.3)ai,j={-242,forj=i,121,for{j=i+1,i=1,2,…,n-1,j=i-1,i=2,3,…,n,0,otherwise.
The exact solution is
(3.4)x={.1915802528sin⁡t-4cos⁡t+3-t2,x<0,-1.462117157e-t-0.5378828428et+1+t2,x>0.
In Figure 1, we compare residual method with Noor et al. [14, 15]. The residual iterative method, minimization method [14], and the iterative method [10] solve (3.1) in 51, 142, and 431 iterations, respectively. For the next two examples, we interchange v1,v2 with each other as Algorithm 2.1 converges for nonzero vectors v1,v2∈Rn.Figure 1Example 3.2 (see [17]).
We first chose a randomA from a uniform distribution on [-10,10], then we chose a random x from a uniform distribution on [−1, 1]. Finally we computed b=Ax-|x|. We ensured that the singular values of each A exceeded 1 by actually computing the minimum singular value and rescaling A by dividing it by the minimum singular value multiplied by a random number in the interval [0, 1]. The computational results are given in Table 1.Table 1Problems with sad(A)>1GNMRIMProblem size1000Number of problem solved100Total number of iterations297268Accuracy10-610-6Total time in seconds870.30977.45In Table1, GNM and RIM denote generalized Newton method [17] and residual iterative method. From Table 1 we conclude that residual method for solving absolute value equations (1.2) is more effective.Example 3.3 (see [23]).
Consider random matrixA and b in Mat lab code as
(3.5)n=input(“dimensionofmatrixA=”);rand(“state”,0);R=rand(n,n);b=rand(n,1);A=R′*Run*eye(n),
with random initial guess. The comparison between the residual iterative method and the Yong method [23] is presented in Table 2.Table 2OrderResidual iterative methodYong method [23]No. of iterationsTOCNo. of iterationsTOC420.00622.230820.02223.3401620.02533.7903220.05324.1206420.07536.69012820.142312.45025620.201334.67051231.436576.570102426.6045157.12In Table2 TOC denotes time taken by CPU. Note that for large problem sizes the residual iterative method converges faster than the Yong method [23].


## 4. Conclusions

In this paper, we have used the projection technique to suggest an iterative method for solving the absolute value equations. The convergence analysis of the proposed method is also discussed. Some examples are given to illustrate the efficiency and implementation of the new iterative method. The extension of the proposed iterative method for solving the general absolute value equation of the formAx+B|x|=b for suitable matrices is an open problem. We have remarked that the variational inequalities are also equivalent to the absolute value equations. This equivalent formulation can be used to suggest and analyze some iterative methods for solving the absolute value equations. It is an interesting and challenging problem to consider the variational inequalities for solving the absolute value equations.



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*Source: 406232-2012-03-07.xml*