Optim Lett (2012) 6:1027–1033 DOI 10.1007/s11590-011-0332-0 SHORT COMMUNICATION
On an iterative method for solving absolute value equations Muhammad Aslam Noor · Javed Iqbal · Khalida Inayat Noor · Eisa Al-Said
Received: 29 September 2010 / Accepted: 15 April 2011 / Published online: 26 May 2011 © Springer-Verlag 2011
Abstract We suggest an iterative method for solving absolute value equation Ax − |x| = b, where A ∈ R n×n is symmetric matrix and b ∈ R n , coupled with the minimization technique. We also discuss the convergence of the proposed method. Some examples are given to illustrate the implementation and efficiency of the method. Keywords Absolute value equations · Positive definite matrix · Minimization technique
1 Introduction We consider absolute value equations of the form Ax − |x| = b,
(1.1)
M. A. Noor (B) · J. Iqbal · K. I. Noor Mathematics Department, COMSATS Institute of Information Technology, Park Road, Islamabad, Pakistan e-mail:
[email protected];
[email protected] J. Iqbal e-mail:
[email protected] K. I. Noor e-mail:
[email protected] M. A. Noor · E. Al-Said Mathematics Department, College of Science, King Saud University, Riyadh, Saudi Arabia E. Al-Said e-mail:
[email protected];
[email protected]
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where A ∈ R n×n is symmetric matrix b ∈ R n and |x| will denote the vector in R n with absolute values of components of x. The absolute value equation (1.1) was investigated in [7–12,16,18,19] and a bilinear program was prescribed there for the special case, when the singular values of A are not less than one. The absolute value equation (1.1) is a special case of the generalized absolute value equation of the type: Ax + B|x| = b,
(1.2)
where B ∈ R n×n is matrix, was introduced and investigated in a more general form in [7,16,18,19]. The significance of the absolute value equation (1.1) arises from the fact that linear programs, quadratic programs, bimatrix games and other problems can all be reduced to an linear complementarity problem (LCP) [3,10]. Mangasarian [7,8] has shown that the absolute value equation (1.1) is equivalent to the linear complementarity problem (LCP). This equivalence formulation has been used by Mangasarian [7,8] to solve the absolute equation using the linear complementarity problems. If B = 0, zero matrix, then generalized absolute value equation (1.2) reduces to a system of linear equations Ax = b, which have several applications in scientific computation, see [1–20]. In this paper, we suggest and analyze an iterative method for solving absolute value equation (1.1) using minimization technique. Our method is simple and easy to implement as compared with linear complementarity problem. Our method examines equations of the absolute value equation (1.1) one at a time in sequence and previously computed results are used as soon as they are available. Let R n be the finite dimension Euclidean space, whose inner product and norm are denoted by ., . and . respectively. For x ∈ R n , 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 is defined as D = δ|x| = diag(sign)(x), the diagonal matrix corresponding to sign(x). We consider the matrix A such that C = A − D is positive definite for any arbitrary matrix D. We note that if A, D are both symmetric matrices, then C is symmetric. We also need the following concept. Definition 1.1 A symmetric matrix A ∈ R n×n is positive definite, if and only if, x, Ax > 0, ∀x ∈ R n . 2 Main results In this section, we discuss our main result which based on minimization technique. For a square matrix A ∈ R n×n and b ∈ R n , we consider the function f (x) = Ax, x − |x|, x − 2b, x, x ∈ R n
(2.1)
Theorem 2.1 If C = A − D is positive definite matrix, then x ∈ R n is a solution of the absolute value equation Ax − |x| = b, if and only if, x ∈ R n is a minimum of the function f (x), where f (x) is defined as (2.1).
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Proof Let x, v ∈ R n and α be a real number variable. Using Taylor’s series, we have f (x + αv) = f (x) + α f (x), v +
α 2 f (x)v, v. 2
(2.2)
Since f (x) = 2(Ax − |x| − b), f (x) = 2(A − D) = 2C. Using these values, we can rewrite (2.2) in the following form f (x + αv) = f (x) + 2αAx − |x| − b, v + α 2 Cv, v.
(2.3)
Consider the function g(α) = f (x) + 2αAx − |x| − b, v + α 2 Cv, v.
(2.4)
It is clear that g(α) has its minimum at α=−
Ax − |x| − b, v , Cv, v > 0. Cv, v
(2.5)
From (2.4) and (2.5), we have Ax − |x| − b, v Ax − |x| − b, v 2 g(α) = f (x) − 2 Ax − |x| − b, v + − Cv, v Cv, v .Cv, v Ax − |x| − b, v2 (2.6) = f (x) − Cv, v So, for any vector v = 0, we have f (x + αv) < f (x). which is impossible. Consequently, we have Ax − |x| − b, v = 0. In this case, it follows that f (x) = f (x + αv). Suppose x ∗ satisfies Ax ∗ − |x ∗ | = b.
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Then Ax ∗ − |x ∗ | − b, v = 0, for any vector v and f (x) cannot be made any smaller than f (x ∗ ). Thus x ∗ minimizes f. On the other hand, suppose that x ∗ minimizes f . Then for any vector v, we have f (x ∗ + αv) ≥ f (x ∗ ). Thus, Ax ∗ − |x ∗ | − b, v = 0. which implies that Ax ∗ − |x ∗ | − b = 0 and consequently, we have Ax ∗ − |x ∗ | = b. This shows that x ∗ is a solution of the absolute value equation (1.1).
Let x0 be an initial approximation and let v1 = 0 be an initial search direction. Theorem 2.1 enables us to suggest the following iterative scheme for solving the absolute value equation (1.1). xk = xk−1 + αk vk ,
(2.7)
where αk = −
Axk−1 − |xk−1 | − b, vk . Cvk , vk
Consider vk = ek , where ek is the kth column of the identity matrix. The iterative scheme (2.7) examines equations of (1.1) one at a time in sequence and previously computed results are used as soon as they are available therefore in this sense this is a Gauss-Seidel method for solving absolute value equation.
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Algorithm 2.1 Choose an initial value x0 ∈ R n , A, C ∈ R n×n , b ∈ R n
Proof of Convergence We now prove the convergence criteria of Algorithm 2.1 and this is the main motivation of our next result. Theorem 2.2 The reduction between f (xk−1 ) and f (xk ) is of equivalence to the reduction of error in the C-norm, where f is the form of (2.2) and C is symmetric. Proof Consider 2 2 − xk−1 − x ∗ C = C xk − C x ∗ , xk − x ∗ − C xk−1 − C x ∗ , xk−1 − x ∗ xk − x ∗ C
= C xk , xk − C xk , x ∗ − C x ∗ , xk + C x ∗ , x ∗ −C xk−1 , xk−1 + C xk−1 , x ∗ + C x ∗ , xk−1
−C x ∗ , x ∗ = C xk , xk − 2C x ∗ , xk − C xk−1 , xk−1 +2C x ∗ , xk−1 Since C is symmetric, so 2 2 − xk−1 − x ∗ C = C xk , xk − 2C x ∗ , xk − [C xk−1 , xk−1 xk − x ∗ C −2C x ∗ , xk−1 ]
= Axk − |xk |, xk − 2Ax ∗ − |x ∗ |, xk −[Axk−1 − |xk−1 |, xk − 2Ax ∗ − |x ∗ |, xk−1 ] = f (xk ) − f (xk−1 ). Hence the result.
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Table 1 Comparison of iterative methods Order
No. of iterations (prob. 1)
No. of iterations (prob. 2)
10 50 100
4 4 5
7 9 8
3 Numerical results To illustrate the implementation and efficiency of the proposed method, we consider the following examples. Example 1 Let A be a matrix whose diagonal elements are 500 and the nondiagonal elements are chosen randomly from the interval [1,2] such that A is asymmetric matrix. Let b = (A − I )e where I is the identity matrix of order n and e is n × 1 vector whose elements are all equal to unity such that x = (1, 1, . . . , 1)T is the exact solution. The stopping criteria is ||xk − xk−1 || < 10−6 and the initial guess is x0 = (0, 0, . . . , 0)T . Example 2 [20] Let the matrix A be given by aii = 4n, ai,i+1 = ai+1,i = n, ai j = 0.5, i = 1, 2, . . . , n. Let b = (A − I )e, where I is the identity matrix order n and e is n × 1 vector whose elements are all equal to unity such that x = (1, 1, . . . , 1)T is the exact solution. The stopping criteria is xk − xk−1 < 10−6 and the initial guess is x0 = (x1 , x2 , . . . , xn )T , xi = 0.001 ∗ i. The numerical results are shown in Table 1. 4 Conclusion In this paper, we have used the minimization technique to suggest and analyze an iterative method for solving the absolute value equation of the type Ax − |x| = b. We have also discussed the convergence criteria of the new iterative method under suitable condition. Examples are given to illustrate the implementation and the efficiency of the proposed iterative method. We would like to mention that the proposed iterative methods can be extended for the generalized absolute value equation of the type Ax + B|x| = b if the matrix A + BD is positive definite matrix for the matrix D = δ|x| = diag(sign)(x), We would like to mention that the variational inequalities are equivalent to the (linear) complementarity problems under some suitable conditions, see [13–16]. This equivalent form may be used to suggest new and novel methods for solving the generalized absolute value equations.
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Acknowledgments This research is supported by the Visiting Professor Program of King Saud University, Riyadh, Saudi Arabia and Research Grant No: KSU.VPP. 108. Authors would like to thank the referees for their very constructive and value comments and Dr. S. M. Junaid Zaidi, Rector, CIIT, Pakistan for providing excellent research facilities.
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