Abstract
In this note, we show that the singular value condition \(\sigma _{\max }(B) < \sigma _{\min }(A)\) leads to the unique solvability of the absolute value equation \(Ax + B|x| = b\) for any b. This result is superior to those appeared in previously published works by Rohn (Optim Lett 3:603–606, 2009).
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1 Introduction
In this note, we consider the following absolute value equation (AVE)
where \(A, B\in \mathbb {R}^{n\times n}\) and \(b\in \mathbb {R}^{n}\). We show that if matrices A and B satisfy
then the AVE (1) for any b has a unique solvability, where \(\sigma _{\max }\) and \(\sigma _{\min }\), respectively, denote the maximal and minimal singular values. This result is weaker than the following condition
which was provided in [1] by Rohn, one can see [1] for more details.
At present, the AVE (1) has attracted considerable attention because the AVE (1) is used as a useful tool in optimization, such as the linear complementarity problem, linear programming and convex quadratic programming, and so on.
Recently, it has been studied from two aspects: one is theoretical analysis, the other is to develop many efficient methods for solving the AVE (1). The former focuses on the theorem of alternatives, various equivalent reformulations, and the existence and nonexistence of solutions, see [1,2,3,4,5,6,7]. Especially, in [6], the authors presented some necessary and sufficient conditions for the unique solution of the AVE (1) with \(B=I\), where I denotes the identity matrix. The later focuses on exploring some numerical methods for solving the AVE (1), such as the smoothing Newton method [8], the generalized Newton method [9], the sign accord method [10], the Picard-HSS method [11], the relaxed nonlinear PHSS-like method [12], Levenberg–Marquardt method [13], the finite succession of linear programs [14], the modified generalized Newton method [15, 16], the preconditioned AOR method [17] and the modified Newton-type method [18].
2 The main result
In this section, we will give our main result.
To give our main result, the following lemma is required.
Lemma 2.1
If matrices A and B satisfy
then the matrix \((A-B)^{-1}(A+B)\) is positive definite.
Proof
Since \(\sigma _{\max }(B) < \sigma _{\min }(A)\), for all nonzero \(x \in {\mathbb {R}}^{n}\), we have
Clearly,
Noting that \(x^{T}BA^{T}x=x^{T}AB^{T}x\). Further, we have
Let \((A^{T}-B^{T})x=y\). By the simple computations, we have
It follows that \(y^{T}(A-B)^{-1}(A+B)y>0\) from Eq. (2), which implies that the matrix \((A-B)^{-1}(A+B)\) is positive definite. \(\square \)
Theorem 2.1
If matrices A and B satisfy
then the AVE (1) for any b has a unique solution.
Proof
Let \(x_{+}=\frac{|x|+x}{2}\) and \(x_{-}=\frac{|x|-x}{2}\). Then
Substituting (3) into (1), we obtain
Based on the results in Lemma 2.1, \((A-B)^{-1}(A+B)\) is a P-matrix. Therefore, the linear complementarity problem (4) is uniquely solvable for any b in [19] and so is the AVE (1) for any b. \(\square \)
On the unique solvability of the AVE (1), the following result was provided in [1].
Theorem 2.2
[1] Let \(A,B\in \mathbb {R}^{n\times n}\) satisfy
then the AVE (1) for any b has a unique solution.
Remark 2.1
It is noted that the condition in Theorem 2.1 may be weaker than the condition in Theorem 2.2. In fact, for \(B\in \mathbb {R}^{n\times n}\), we have
Based on Theorem 8.1.18 in [20], we have
where \(\rho (\cdot )\) denotes the spectral radius of the matrix. It follows that \(\sigma _{\max }(B)\le \sigma _{\max }(|B|)\).
Remark 2.2
When \(B=I\), Theorem 2.1 reduces to Proposition 3 (i) in [2]. That is to say, Theorem 2.1 is a generalization of Proposition 3 (i) in [2].
The following example shows that Theorem 2.2 in [1] may be invalid to judge the unique solution of the certain AVE, whereas, Theorem 2.1 can judge the unique solution of the certain AVE.
Example 2.1
Consider the following AVE
By the simple computations, \(\sigma _{\min }(A)=\sigma _{\max }(A)=2.5\), \(\sigma _{\max }(B)=2.3028\) and \(\sigma _{\max }(|B|)=2.618\). When we use Theorem 2.2 in [1], we find that \(\sigma _{\min }(A)<\sigma _{\max }(|B|)\). Clearly, Theorem 2.2 is invalid to judge the unique solution of the AVE (5). Whereas, when using Theorem 2.1, i.e., \(\sigma _{\min }(A)>\sigma _{\max }(B)\), we find that the AVE (5) is unique solution. In fact, the unique solution of the AVE (5) is \(x_{1}=x_{2}=1\). This further shows that Theorem 2.1 is indeed superior to Theorem 2.2 in [1] under certain conditions.
References
Rohn, J.: On unique solvability of the absolute value equation. Optim. Lett. 3, 603–606 (2009)
Mangasarian, O.L., Meyer, R.R.: Absolute value equations. Linear Algebra Appl. 419, 359–367 (2006)
Rohn, J., Hooshyarbakhsh, V., Farhadsefat, R.: An iterative method for solving absolute value equations and sufficient conditions for unique solvability. Optim. Lett. 8, 35–44 (2014)
Zhang, C., Wei, Q.-J.: Global and finite convergence of a generalized newton method for absolute value equations. J. Optim. Theory Appl. 143, 391–403 (2009)
Wu, S.-L., Guo, P.: On the unique solvability of the absolute value equation. J. Optim. Theory Appl. 169, 705–712 (2016)
Wu, S.-L., Li, C.-X.: The unique solution of the absolute value equations. Appl. Math. Lett. 76, 195–200 (2018)
Hladík, M.: Bounds for the solutions of absolute value equations. Comput. Optim. Appl. 69(1), 243–266 (2018)
Caccetta, L., Qu, B., Zhou, G.-L.: A globally and quadratically convergent method for absolute value equations. Comput. Optim. Appl. 48, 45–58 (2011)
Mangasarian, O.L.: A generalized Newton method for absolute value equations. Optim. Lett. 3, 101–108 (2009)
Rohn, J.: An algorithm for solving the absolute value equations. Electron. J. Linear Algebra. 18, 589–599 (2009)
Salkuyeh, D.K.: The Picard-HSS iteration method for absolute value equations. Optim. Lett. 8, 2191–2202 (2014)
Zhang, J.-J.: The relaxed nonlinear PHSS-like iteration method for absolute value equations. Appl. Math. Comput. 265, 266–274 (2015)
Iqbal, J., Iqbal, A., Arif, M.: Levenberg–Marquardt method for solving systems of absolute value equations. J. Comput. Appl. Math. 282, 134–138 (2015)
Mangasarian, O.L.: Absolute value equation solution via concave minimization. Optim. Lett. 1, 3–8 (2007)
Li, C.-X.: A modified generalized Newton method for absolute value equations. J. Optim. Theory Appl. 170, 1055–1059 (2016)
Lian, Y.-Y., Li, C.-X., Wu, S.-L.: Weaker convergent results of the generalized Newton method for the generalized absolute value equations. J. Comput. Appl. Math. 338, 221–226 (2018)
Li, C.-X.: A preconditioned AOR iterative method for the absolute value equations. Int. J. Comput. Methods 14, 1750016 (2017)
Wang, A., Cao, Y., Chen, J.-X.: Modified Newton-type iteration methods for generalized absolute value equations. J. Optim. Theory Appl. 181, 216–230 (2019)
Rohn, J.: A theorem of the alternatives for the equation \(Ax + B|x| = b\). Linear Multilinear A. 52, 421–426 (2004)
Horn, R.A., Johnson, C.R.: Matrix Analysis. Cambridge University Press, Cambridge (1985)
Acknowledgements
The authors would like to thank the anonymous referees for providing helpful suggestions, which greatly improved the paper. Funding was provided by National Natural Science Foundation of China (No. 11961082) and 17HASTIT012.
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Wu, SL., Li, CX. A note on unique solvability of the absolute value equation. Optim Lett 14, 1957–1960 (2020). https://doi.org/10.1007/s11590-019-01478-x
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DOI: https://doi.org/10.1007/s11590-019-01478-x