- Research
- Open Access
A new two-step iterative method for solving absolute value equations
- Jingmei Feng^{1, 2}Email author and
- Sanyang Liu^{1}
https://doi.org/10.1186/s13660-019-1969-y
© The Author(s) 2019
- Received: 20 September 2018
- Accepted: 15 January 2019
- Published: 19 February 2019
Abstract
We describe a new two-step iterative method for solving the absolute value equations \(Ax-|x|=b\), which is an NP-hard problem. This method is globally convergent under suitable assumptions. Numerical examples are given to demonstrate the effectiveness of the method.
Keywords
- Absolute value equations
- Iterative method
- Global convergence
1 Introduction
According to the advantages and disadvantages of the above algorithms, we will present a two-step iterative method for effectively solving AVEs (1). Firstly, we present a new iterative formula, which absorbs the advantages of both the classic Newton and the two-step Traub iterative formulas, it has the characteristics of fast iteration and good convergence. In addition, we incorporate the good idea of solving 1-dimensional nonlinear equations to our iterative formula. Then a new algorithm for solving AVEs (1) is designed, we can prove that this method can converge to the global optimal solution of AVEs (1). Finally, numerical results and comparison with the classic Newton and two-step Traub iterative formulas show that our method converges faster and the solution accuracy is higher.
This article is arranged as follows. Section 2 is the preliminary about AVEs (1). We give a new two-step iterative method and prove the global convergence of the proposed method in Sect. 3. In Sect. 4 we present the numerical experiments. Some concluding remarks to end the paper in Sect. 5.
2 Preliminaries
By calculating 100 randomly generated 1000-dimensional AVEs (1), Mangasarian proved that this Newton iteration is an effective method, when the singular values of A are not less than 1. The vector iterations \(\{x^{k}\}\) linearly converge to the true solution of AVEs (1),
Although the computation time of the iterative formula (4) is greater than that of (5), the experiment’s results obtained by the iterative formula (4) are better than that of (5).
Some iterative methods with higher order convergence and high precision for solving nonlinear equations \(g(x)=0\), where \(g:D\subset R\rightarrow R\), are in [18], which give us some inspiration and motivate us to extend those methods to the n-dimensional problem, especially the high-dimensional absolute value equations. Combining with the above-mentioned methods, we designed the following effective methods.
3 Algorithm and convergence
Based on the iterative formula (6), we design Algorithm 3.1 for solving AVEs (1).
Algorithm 3.1
Next, we prove the global convergence of Algorithm 3.1.
Lemma 3.1
The singular values of the matrix \(A\in R^{n\times n}\) exceed 1 if and only if the minimum eigenvalue of \(A'A\) exceeds 1.
Proof
See [5]. □
Lemma 3.2
If all the singular values of \(A\in R^{n\times n}\) exceed 1 for the method (6), then, for any diagonal matrix D, whose diagonal elements \(D_{ii}=+1, 0, -1\), \(i=1, 2,\ldots, n\). \((A-D)^{-1}\) exists.
Proof
Hence, \((A-D)\) is nonsingular, and the sequence \(\{x^{k}\}\) produced by (6) is well defined for any initial vector \(x_{0} \in R ^{n}\). This proof is similar to [16]. □
Lemma 3.3
(Lipschitz continuity of the absolute value)
Proof
The proof follows the lines of [16].
Lemma 3.4
If the singular values of symmetric matrix \(A\in R^{n\times n}\) exceed 1, then the direction \(d^{k}_{2}\) of (9) is a descent direction for the objective function \(F(x)\), where \(F(x)=\frac{1}{2} \Vert f(x)\Vert ^{2}\).
Proof
Since \(f(x)=Ax-\vert x\vert -b\), \(f'(x)=\partial f(x)=A-D(x)\), \((A-D(x))^{-1}\) exists for any diagonal matrix D, whose diagonal elements \(D_{ii}=+1, 0, -1\), \(i=1, 2,\ldots, n\), and \((f'(x))^{T}=f'(x)\), \(F(x)=\frac{1}{2}\Vert f(x)\Vert ^{2}\), \(F'(x)=f'(x)f(x)\).
Thus, \(d^{k}_{1}\) is not a descent direction of \(F(x)\), we have \(\Vert f(y^{k})\Vert >\Vert f(x^{k})\Vert \).
Theorem 3.1
(Global convergence)
If the norm of the symmetric matrix A exists, then the norm of \(A-D\) is existent for any diagonal matrix D whose diagonal elements are ±1 or o. Consequently, the sequence \(\{x^{k}\}\) is a Cauchy series, thus \(\{x^{k}\}\) converges to the unique solution x̄ of AVEs (1).
Proof
For \(\forall k \in Z\), \(f(x^{k})=Ax^{k}-\vert x^{k}\vert -b\), \(f(y ^{k})=Ay^{k}-\vert y^{k}\vert -b\).
4 Numerical results
In this section we consider some examples to illustrate the feasibility and effectiveness of Algorithm 3.1. All the experiments are performed by Matlab R2010a. We compare the proposed method (TSI) with the generalized Newton method (4) (GNM) and the generalized Traub method (5) (GTM).
Example 1
Example 2
The comparison of GNM, GTM and TSI in Example 1
Dim | GNM | GTM | TSI | ||||||
---|---|---|---|---|---|---|---|---|---|
K | ACC | T | K | ACC | T | K | ACC | T | |
100 | 3 | 6.6601 × 10^{−12} | 0.0036 | 3 | 7.3677 × 10^{−13} | 0.0069 | 3 | 4.4030 × 10^{−13} | 0.0122 |
200 | 3 | 1.0395 × 10^{−10} | 0.0145 | 3 | 2.2623 × 10^{−12} | 0.0121 | 3 | 1.0761 × 10^{−12} | 0.0313 |
300 | 3 | 2.9896 × 10^{−10} | 0.0432 | 3 | 2.8670 × 10^{−12} | 0.0611 | 3 | 1.2862 × 10^{−12} | 0.0943 |
400 | 3 | 6.1301 × 10^{−10} | 0.8682 | 3 | 5.8058 × 10^{−12} | 0.0075 | 3 | 1.4559 × 10^{−12} | 0.1137 |
500 | 3 | 1.3574 × 10^{−10} | 0.1498 | 3 | 6.1348 × 10^{−12} | 0.1355 | 3 | 1.6708 × 10^{−12} | 0.1850 |
600 | 3 | 4.3000 × 10^{−9} | 0.2231 | 3 | 7.8633 × 10^{−12} | 0.2702 | 3 | 3.6379 × 10^{−12} | 0.2897 |
700 | 3 | 1.2489 × 10^{−9} | 0.5052 | 3 | 7.8369 × 10^{−12} | 0.2932 | 3 | 4.2171 × 10^{−12} | 0.4356 |
800 | 3 | 3.9699 × 10^{−9} | 0.8166 | 4 | 1.6596 × 10^{−11} | 0.4496 | 3 | 4.3618 × 10^{−12} | 0.6221 |
900 | 3 | 5.7449 × 10^{−9} | 1.1213 | 4 | 1.7759 × 10^{−11} | 0.7850 | 3 | 4.5474 × 10^{−12} | 0.8392 |
1000 | 3 | 6.5951 × 10^{−9} | 1.5295 | 4 | 1.7845 × 10^{−11} | 0.9808 | 3 | 5.4310 × 10^{−12} | 1.0813 |
The comparison of GNM, GTM and TSI in Example 2
Dim | GNM | GTM | TSI | ||||||
---|---|---|---|---|---|---|---|---|---|
K | ACC | T | K | ACC | T | K | ACC | T | |
100 | 3 | 5.8699 × 10^{−12} | 0.0052 | 3 | 3.0920 × 10^{−13} | 0.0080 | 3 | 3.0515 × 10^{−13} | 0.0064 |
200 | 3 | 8.7417 × 10^{−11} | 0.0171 | 3 | 8.5750 × 10^{−13} | 0.0175 | 3 | 8.3127 × 10^{−13} | 0.0204 |
300 | 3 | 2.0700 × 10^{−10} | 0.0508 | 3 | 1.5518 × 10^{−12} | 0.0578 | 3 | 1.6336 × 10^{−12} | 0.0467 |
400 | 3 | 4.2251 × 10^{−10} | 0.1011 | 3 | 2.5566 × 10^{−12} | 0.1143 | 3 | 2.5549 × 10^{−12} | 0.1314 |
500 | 4 | 5.2788 × 10^{−10} | 0.1959 | 4 | 3.5830 × 10^{−12} | 0.2092 | 4 | 3.5426 × 10^{−12} | 0.2462 |
600 | 4 | 3.0194 × 10^{−10} | 0.3553 | 4 | 4.7330 × 10^{−12} | 0.3736 | 4 | 4.7270 × 10^{−12} | 0.5042 |
700 | 4 | 7.0378 × 10^{−10} | 0.5615 | 4 | 6.0018 × 10^{−12} | 0.5323 | 4 | 5.8574 × 10^{−12} | 0.7583 |
800 | 4 | 1.0602 × 10^{−10} | 0.8279 | 5 | 7.6437 × 10^{−12} | 0.8035 | 5 | 6.7769 × 10^{−12} | 0.8134 |
900 | 5 | 2.8031 × 10^{−9} | 1.6159 | 5 | 8.8566 × 10^{−12} | 1.4304 | 5 | 8.4667 × 10^{−12} | 1.4072 |
1000 | 5 | 3.5603 × 10^{−9} | 2.3006 | 5 | 1.0776 × 10^{−11} | 1.9268 | 5 | 9.9720 × 10^{−12} | 1.8344 |
In Tables 1 and 2, Dim, K, ACC and T denote the dimension of the problem, the number of iterations, \(\Vert Ax^{k}-\vert x^{k}\vert -b\Vert _{2}\) and time(s), respectively. It is clear from Tables 1 and 2 that the new two-step iterative method is very effective in solving absolute value equations, especially the high dimension problem.
5 Conclusions
In this paper, we propose a new two-step iterative method for solving non-differentiable and NP-hard absolute value equations \(Ax-\vert x\vert =b\), when the minimum singular value of A is greater than 1. Compared with the existing methods GNM and GTM, our new method has some nice convergence properties and better calculation consequences. In the future, we have the confidence to continue an in-depth study.
Declarations
Acknowledgements
The authors are very grateful to the editors and referees for their constructive advice.
Funding
The research are supported by the National Natural Science Foundation of China (Grant No. 61877046); the second batch of young outstanding talents support plan of Shaanxi universities.
Authors’ contributions
All authors contributed equally to the manuscript, and they read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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