A note on unique solvability of the absolute value equation

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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References

  1. 1.

    Rohn, J.: On unique solvability of the absolute value equation. Optim. Lett. 3, 603–606 (2009)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Mangasarian, O.L., Meyer, R.R.: Absolute value equations. Linear Algebra Appl. 419, 359–367 (2006)

    MathSciNet  Article  Google Scholar 

  3. 3.

    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)

    MathSciNet  Article  Google Scholar 

  4. 4.

    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)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Wu, S.-L., Guo, P.: On the unique solvability of the absolute value equation. J. Optim. Theory Appl. 169, 705–712 (2016)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Wu, S.-L., Li, C.-X.: The unique solution of the absolute value equations. Appl. Math. Lett. 76, 195–200 (2018)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Hladík, M.: Bounds for the solutions of absolute value equations. Comput. Optim. Appl. 69(1), 243–266 (2018)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Caccetta, L., Qu, B., Zhou, G.-L.: A globally and quadratically convergent method for absolute value equations. Comput. Optim. Appl. 48, 45–58 (2011)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Mangasarian, O.L.: A generalized Newton method for absolute value equations. Optim. Lett. 3, 101–108 (2009)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Rohn, J.: An algorithm for solving the absolute value equations. Electron. J. Linear Algebra. 18, 589–599 (2009)

    MathSciNet  MATH  Google Scholar 

  11. 11.

    Salkuyeh, D.K.: The Picard-HSS iteration method for absolute value equations. Optim. Lett. 8, 2191–2202 (2014)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Zhang, J.-J.: The relaxed nonlinear PHSS-like iteration method for absolute value equations. Appl. Math. Comput. 265, 266–274 (2015)

    MathSciNet  MATH  Google Scholar 

  13. 13.

    Iqbal, J., Iqbal, A., Arif, M.: Levenberg–Marquardt method for solving systems of absolute value equations. J. Comput. Appl. Math. 282, 134–138 (2015)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Mangasarian, O.L.: Absolute value equation solution via concave minimization. Optim. Lett. 1, 3–8 (2007)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Li, C.-X.: A modified generalized Newton method for absolute value equations. J. Optim. Theory Appl. 170, 1055–1059 (2016)

    MathSciNet  Article  Google Scholar 

  16. 16.

    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)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Li, C.-X.: A preconditioned AOR iterative method for the absolute value equations. Int. J. Comput. Methods 14, 1750016 (2017)

    MathSciNet  Article  Google Scholar 

  18. 18.

    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)

    MathSciNet  Article  Google Scholar 

  19. 19.

    Rohn, J.: A theorem of the alternatives for the equation \(Ax + B|x| = b\). Linear Multilinear A. 52, 421–426 (2004)

    MathSciNet  Article  Google Scholar 

  20. 20.

    Horn, R.A., Johnson, C.R.: Matrix Analysis. Cambridge University Press, Cambridge (1985)

    Google Scholar 

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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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Correspondence to Shi-Liang Wu.

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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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Keywords

  • Absolute value equation
  • Unique solution
  • Singular values