An improved generalized Newton method for absolute value equations

In this paper, we suggest and analyze an improved generalized Newton method for solving the NP-hard absolute value equations \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Ax-|x|=b$$\end{document}Ax-|x|=b when the singular values of A exceed 1. We show that the global and local quadratic convergence of the proposed method. Numerical experiments show the efficiency of the method and the high accuracy of calculation.

assumption and gave some existence and convexity results for the solution set of the AVEs (1) in Hu and Huang (2010). Paper Zhang et al. (2015) presented a new algorithm which relaxes the absolute value equations into a convex optimization problem, Zhang et al. found the sparsest solution of AVEs by the minimum l 0 -norm. Caccetta et al. proposed a globally and quadratically convergent method for AVEs in Caccetta et al. (2011). Rohn et al. gave an iterative method for AVEs (1) and analyzed the sufficient conditions of unique solvability by Rohn et al. (2014), Uniqueness is always a hot spot (Wu and Guo 2015), and Moosaei et al. gave the minimum norm solution of absolute value equations Ax − |x| = b which has multiple solutions (at most 2 n ) via simulated annealing algorithm in Moosaei et al. (2014). Salkuyeh (2014) put forward a hybrid algorithm which combined with skew-Hermitian and Picard-Hermitian splitting iteration method for solving AVEs (1), and gave the convergence analysis of the hybrid algorithm. Furthermore, Mangasarian (2009) clearly showed that generalized Newton method is a very effective method by solving some high dimensional examples in very few iterations. Haghani (2015) proposed an improved Newton method with two-step form, called Traub's method, whose effectiveness is better than that of Mangasarian's. Iqbal et al. (2015) proposed a Levenberg-Marquardt method for AVEs (1), which is the combination of steepest descent and the Gauss-Newton method. Paper Cruz et al. (2016) raised an inexact semi-smooth Newton algorithm for AVEs (1), and proved that the method is globally convergent. So, how to solve absolute value equations based on the classic Newton algorithm have been received many of the concerns. That have motivated us for trying to improve Newton method as the main aim of the present paper. Now we describe our notation. The scalar product of two vectors x and y in the n-dimensional real space will be denoted by x, y . For x ∈ R n , the norm x will denote the two-norm (x T x) 1/2 , and sign(x) will denote a vector with components equal to +1, 0 or −1, depending on whether the corresponding component of x is positive, zero or negative, respectively. In addition, diag(sign(x)) will denote a diagonal matrix corresponding to sign (x).
In "Preliminary" section of the present work we give the notations and preliminary notes about AVEs. "The improved generalized Newton method" section gives an improved generalized Newton iterative algorithm which is globally and locally quadratic convergent under certain assumptions. In "Computational results" section some numerical reports show the efficiency of the proposed scheme. "Conclusions" section gives some concluding remarks to end the paper.

Preliminary
We begin by defining the piece-wise linear vector function f(x) specified by the AVEs (1) as follows: A generalized Jacobian of f at x is where D(x) = ∂|x| = diag(sign(x)). ( To solve (1), the iterative computational method of Mangasarian (2009) is as follows: Another method proposed by Haghani (2015) is as follows: It has been shown that both the sequences {x k } generated by Eqs. (4) and (5) converge linearly to the true solution x of AVEs (1) when the singular values of A are exceed 1. However, the computational time of (5) is a little less than that of (4), with a higher residual error precision than (4). Ostrowski (1960) and Traub (1964) presented a modified Newton's iteration for solving nonlinear equation in real space R, which give us some inspiration. We will promote this idea to the n-dimensional space. The iterative method is as follows:

The improved generalized Newton method
We can simplify (6) as much as possible to obtain the follwing form of improved generalized Newton method for AVEs . It is clear that our method is a Newton method with a specific liner search.

Algorithm 1 An improved generalized Newton method
Step 0. Choose an initial vector x ∈ R n to AVE (1), set k = 0. Step Step 3. Set k = k + 1, go to Step 1.
We shall need a few theoretical results to establish convergence of Algorithm 1, we first quote the following two results from Mangasarian and Meyer (2006).

Lemma 2
If the singular values of A ∈ R n×n exceed 1 for the method (6), then (A − D) −1 exists for any diagonal matrix D whose diagonal elements are ±1 or 0.
Therefore, the sequence of vector iterates form (6) is well defined no matter how to choose the initial vector x 0 in R n .
We now prove the proposed Newton direction d k of (7) is a descent direction for the objective function f (x) 2 .

Lemma 3 If the singular values of
, and (A − D(x)) −1 exists for any diagonal matrix D whose diagonal elements are ±1 or 0.
In addition, we know and Moreover,

Lemma 4 Let the singular values of A exceed 1, then the sequence {x k } generated by the improved generalized Newton method (6) is bounded, and there exists an accumulation
where D is a assured diagonal matrix with diagonal elements equal to ±1 or 0 extracted from the finite number of possible configurations for D(x k ) in the sequence {D(x k )}, and such that the bounded subsequence x k j +1 �x k j +1 � converges to x. By Eq. (6), we have Due to the following equation: and Lemma 3, the Newton direction d k in (7) is a descent direction for the objective function f (x) 2 . We get �a k j · f (x k j )� → 0, as j → ∞. Now, j → ∞ gives us: since x k j +1 → ∞. This is a contradiction with the nonsingularity of (A − D) which follows from Lemma 2. Hence, the vector sequence {x k } is bounded and there exists an The proof is complete.
Theorem 1 (Global linear convergence) If �(A − D) −1 � < 1 3 for any diagonal matrix D with diagonal elements of ±1 or 0, then the improved generalized Newton method (6) converges linearly from any starting point x 0 to a solution x for any solvable AVEs (1).
Theorem 2 If �A −1 � < 1 4 and D(x k ) � = 0 for any diagonal matrix D(x k ) with diagonal elements of ±1 or 0, then the Algorithm 1 converges linearly from any starting point x 0 to a solution x for any solvable AVEs (1).
Proof The proof directly from Mangasarian (2009, Proposition 7). It is hence omitted.
In the following, we use Eq. (7) to prove the locally quadratic convergence of the Algorithm 1. Lemma 5 If A − D is nonsingular for any diagonal matrix D with diagonal elements of ±1 or 0, then the Algorithm 1 is approximately Newton's method.
Proof Taking into account the step length When x k is in a neighborhood of the solution x of AVEs (1), f (y k ) is close to f (x k ), we perform a Newton step from x k to y k , for larger k, f (y k ) = O(�y k −x� 2 ), thus Hence, our method is approximately the generalized Newton's method.
Theorem 3 (Locally quadratic convergence) If A − D is nonsingular for any diagonal matrix D with diagonal elements of ±1 or 0, then the sequence {x k } from improved generalized Newton's method (7) converges to x and �x k+1 −x� = O(�x k −x� 2 ).
Proof This theorem can be proved in a similar way as Theorem 2 by Qi and Sun (1993). We omit it here.

Computational results
To illustrate the implementation and efficiency of the proposed method, we test the following two examples. All the experiments are performed by MATLAB R2010a. In comparisons, NM, TM and INM denote the generalized Newton's method (4), the generalized Traub's method (5) and the improved generalized Newton's method (6), respectively.
Example 1 We choose randomly matrix A according to the following formula: Then, we choose a random x ∈ R n , compute b = Ax − |x|, and denote the problem size by n. To ensure that the minimum singular value of each A exceeds 1, we compute the minimum singular value of A and reseal A by the minimum singular value multiply by a random number in the interval [1,2], The results are shown in Table 1.
A = rand(n * (eye(n, n) − 0.02 * (2 * rand(n, n) − 1))). Example 2 The matrix A of which all the singular values are greater than 1 is generated by the following MATLAB procedure: And, the results are shown in Table 2.
In Tables 1 and 2, Dim, K, ACC and T denote the dimensions of the problem, the number of iterations, �Ax k − |x k | − b� 2 and times(s), respectively. It is evident from Tables 1 and 2 that the improved generalized Newton's method is very effective for solving large problems.
rand('state', 0); R = rand(n, n); A = R ′ * R + n * eye(n); b = (A − eye(n, n)) * ones(n, 1). We give below the convergence curves of three algorithms for solving Examples 1, 2 by Figs. 1 and 2. We can see that the convergence of the INM is better than NM's and TM's.

Conclusions
In this paper, we have proposed the generalized Newton's method with special search direction for solving the NP-hard absolute value equations under certain assumptions on A. The method have some nice convergence properties and calculation results. Further work is to find more effective methods for AVEs.