We suggest and analyze a residual iterative method for solving absolute value equations Ax-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.
1. Introduction
The residual methods were proposed for solving large sparse systems of linear equations
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:
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 by K 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,
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 asD(x)=∂|x|=diag(sign(x)),
where ∂|x| represent the generalized Jacobean of |x| based on a subgradient [20, 21].
We denote the following by
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:
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 subspaceK1=span{v1},L1=span{Cv1},x0=xk.
For D(x̃k+1)=D(xk), we write the residual in the following form:
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
x̃k+1∈xk+K1suchthatb-Cx̃k+1⊥L1;
that is, we find the approximate solution by the iterative scheme
x̃k+1=xk+αv1.
Now, we rewrite (2.4) in the inner product as
〈b-Cx̃k+1,Cv1〉=0;
from the above discussion, we have
〈b-Cxk-αCv1,Cv1〉=〈b-Cxk,Cv1〉-α〈Cv1,Cv1〉=p1-aα=0,
from which we have
α=p1a.
The next step is to choose the subspace
K2=span{v2},L2=span{Cv2},x0=x̃k+1,
and to find the approximate solution xk+1 such that
xk+1∈x̃k+1+K2suchthatb-Cxk+1⊥L2,
where
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〈b-Cxk+1,Cv2〉=0.
Thus, we have〈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 obtainβ=ap2-cp1ad.
We remark that one can choose v1=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 guess x0∈Rn,
For k=0,1,2,… until convergence do
rk=b-Axk+|xk|
gk=(A-D(xk))T(Axk-|xk|-b)
Hk=((A-D(xk))-1(A-D(xk)))T
sk=-Hkgk
If ∥rk∥=0, then stop; else
αk=p1/a,βk=(ap2-cp1)/ad
Set xk+1=xk+αkrk+βksk
If ∥xk+1-xk∥<10-6
then stop
End if
End for k.
If β=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
‖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
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
‖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
‖rk‖2-‖rk+1‖2=p12a+(ap2-cp1)2a2d,
the required result (2.15).
Since p12/a+(ap2-cp1)2/a2d≥0, so from (2.18) we have
‖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 Algorithm 2.1, and it is the motivation of our next result.
Theorem 2.3.
If C 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
‖rk‖2-‖rk+1‖2≥p12a=〈rk,Crk〉2〈Crk,Crk〉≥λmin2‖rk‖4λmax2‖rk‖2=λmin2λmax2‖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:
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:
Ax-|x|=b,
where the system matrix A of size n=10 is given by
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
x={.1915802528sint-4cost+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.
Example 3.2 (see [17]).
We first chose a random A 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.
Problems with sad (A)>1
GNM
RIM
Problem size
1000
Number of problem solved
100
Total number of iterations
297
268
Accuracy
10-6
10-6
Total time in seconds
870.30
977.45
In Table 1, 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 matrix A and b in Mat lab code as
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.
Order
Residual iterative method
Yong method [23]
No. of iterations
TOC
No. of iterations
TOC
4
2
0.006
2
2.230
8
2
0.022
2
3.340
16
2
0.025
3
3.790
32
2
0.053
2
4.120
64
2
0.075
3
6.690
128
2
0.142
3
12.450
256
2
0.201
3
34.670
512
3
1.436
5
76.570
1024
2
6.604
5
157.12
In Table 2 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 form Ax+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.
Acknowledgments
This research is supported by the Visiting Professor Program of the King Saud University, Riyadh, Saudi Arabia, and Research Grant no. KSU.VPP.108. The authors are also grateful to Dr. S. M. Junaid Zaidi, Rector, COMSATS Institute of Information Technology, Pakistan, for providing the excellent research facilities.
PaigeC. C.SaundersM. A.Solutions of sparse indefinite systems of linear equations1975124617629038371510.1137/0712047SaadY.SchultzM. H.GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems1983254Yale UniversityAxelssonO.Conjugate gradient type methods for unsymmetric and inconsistent systems of linear equations19802911610.1016/0024-3795(80)90226-8562745ZBL0439.65020JeaK. C.YoungD. M.Generalized conjugate-gradient acceleration of nonsymmetrizable iterative methods19803415919410.1016/0024-3795(80)90165-2591431ZBL0463.65025SaadY.Krylov subspace methods for solving large unsymmetric linear systems19813715510512610.2307/2007504616364ZBL0474.65019MangasarianO. L.MeyerR. R.Absolute value equations20064192-335936710.1016/j.laa.2006.05.0042277975ZBL1172.15302MangasarianO. L.Absolute value programming2007361435310.1007/s10589-006-0395-52292292MangasarianO. L.Absolute value equation solution via concave minimization2007113810.1007/s11590-006-0005-62357603ZBL1149.90098NoorM. A.General variational inequalities19881211912210.1016/0893-9659(88)90054-7953368ZBL0655.49005Aslam NoorM. A.Some developments in general variational inequalities2004152119927710.1016/S0096-3003(03)00558-72050063ZBL1134.49304NoorM. A.Extended general variational inequalities200922218218610.1016/j.aml.2008.03.0072482273ZBL1163.49303NoorM. A.NoorK. I.RassiasT. M.Some aspects of variational inequalities199347328531210.1016/0377-0427(93)90058-J1251844ZBL0788.65074 NoorM. A.NoorK. I.Al-SaidE.Iterative methods for solving nonconvex equilibrium problems2012616569NoorM. A.IqbalJ.KhattriS.Al-SaidE.A new iterative method for solving absolute value equations2011617931797NoorM. A.IqbalJ.NoorK. I.Al-SaidE.On an iterative method for solving absolute value equationsOptimization Letters. In press10.1007/S11590-011-0332-0JingY.-F.HuangT.-Z.On a new iterative method for solving linear systems and comparison results20082201-2748410.1016/j.cam.2007.07.0352444155ZBL1159.65037MangasarianO. L.A generalized Newton method for absolute value equations20093110110810.1007/s11590-008-0094-52453508ZBL1154.90599MangasarianO. L.Solution of symmetric linear complementarity problems by iterative methods1977224465485045883110.1007/BF01268170ZBL0341.65049MangasarianO. L.The linear complementarity problem as a separable bilinear program19956215316110.1007/BF010967651322877ZBL0835.90102RockafellarR. T.New applications of duality in convex programmingProceedings of the 4th Conference On Probability1971Brasov, RomaniaRohnJ.A theorem of the alternatives for the equation Ax+Bx=b200452642142610.1080/03081080420002206862102197ZBL1070.15002SaadY.19962ndBoston, Mass, USAThe PWSYongL.yonglongquan@sohu.comParticle Swarm Optimization for absolute value equations20106723592366