Skip to main content
Springer Nature Link
Log in

Optimal correction of the absolute value equations

  • Published:
Journal of Global Optimization Aims and scope Submit manuscript

Abstract

In this paper, we study the optimum correction of the absolute value equations through making minimal changes in the coefficient matrix and the right hand side vector and using spectral norm. This problem can be formulated as a non-differentiable, non-convex and unconstrained fractional quadratic programming problem. The regularized least squares is applied for stabilizing the solution of the fractional problem. The regularized problem is reduced to a unimodal single variable minimization problem and to solve it a bisection algorithm is proposed. The main difficulty of the algorithm is a complicated constraint optimization problem, for which two novel methods are suggested. We also present optimality conditions and bounds for the norm of the optimal solutions. Numerical experiments are given to demonstrate the effectiveness of suggested methods.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

References

  1. Amaral, P., Barahona, P.: Connections between the total least squares and the correction of an infeasible system of linear inequalities. Linear Algebra Appl. 395, 191–210 (2005)

    MathSciNet  MATH  Google Scholar 

  2. Amaral, P., Barahona, P.: A framework for optimal correction of inconsistent linear constraints. Constraints 10(1), 67–86 (2005)

    MathSciNet  MATH  Google Scholar 

  3. Amaral, P., Fernandes, L.M., Júdice, J., Sherali, H.D.: On optimal zero-preserving corrections for inconsistent linear systems. J. Global Optim. 45(4), 645–666 (2009)

    MathSciNet  MATH  Google Scholar 

  4. Beck, A., Ben-Tal, A.: On the solution of the Tikhonov regularization of the total least squares problem. SIAM J. Optim. 17(1), 98–118 (2006)

    MathSciNet  MATH  Google Scholar 

  5. Bertsekas, D.: Nonlinear Programming, 2nd edn. Athena Scientific, Belmont (1999)

    MATH  Google Scholar 

  6. Björck, A.: Numerical Methods for Least Squares Problems. SIAM, Philadelphia (1996)

    MATH  Google Scholar 

  7. Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge University Press, Cambridge (2004)

    MATH  Google Scholar 

  8. Cottle, R.W., Dantzig, G.B.: Complementary pivot theory of mathematical programming. Linear Algebra Appl. 1(1), 103–125 (1968)

    MathSciNet  MATH  Google Scholar 

  9. Cottle, R.W., Pang, J.S., Stone, R.E.: The Linear Complementarity Problem. Academic Press, Boston (1992)

    MATH  Google Scholar 

  10. David, G.L., Ye, Y.: Linear and Nonlinear Programming. Springer, New York (2008)

    MATH  Google Scholar 

  11. Gauvin, J., Dubeau, F.: Differential properties of the marginal function in mathematical programming. In: Guignard, M. (ed.) Optimality and Stability in Mathematical Programming, Mathematical Programming Studies, vol. 19, pp. 101–119. Springer, Berlin (1982)

    MATH  Google Scholar 

  12. Golub, G.H., Hansen, P.C., O’Leary, D.P.: Tikhonov regularization and total least squares. SIAM J. Matrix Anal. Appl. 21(1), 185–194 (1999)

    MathSciNet  MATH  Google Scholar 

  13. Higham, N.J.: Accuracy and Stability of Numerical Algorithms. SIAM, Philadelphia (2002)

    MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  15. Hladík, M., Rohn, J.: Radii of solvability and unsolvability of linear systems. Linear Algebra Appl. 503, 120–134 (2016)

    MathSciNet  MATH  Google Scholar 

  16. Hu, S.L., Huang, Z.H.: A note on absolute value equations. Optim. Lett. 4(3), 417–424 (2010)

    MathSciNet  MATH  Google Scholar 

  17. 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  MATH  Google Scholar 

  18. Ketabchi, S., Moosaei, H.: An efficient method for optimal correcting of absolute value equations by minimal changes in the right hand side. Comput. Math. Appl. 64(6), 1882–1885 (2012)

    MathSciNet  MATH  Google Scholar 

  19. Ketabchi, S., Moosaei, H.: Minimum norm solution to the absolute value equation in the convex case. J. Optim. Theory Appl. 154(3), 1080–1087 (2012)

    MathSciNet  MATH  Google Scholar 

  20. Ketabchi, S., Moosaei, H.: Optimal error correction and methods of feasible directions. J. Optim. Theory Appl. 154(1), 209–216 (2012)

    MathSciNet  MATH  Google Scholar 

  21. Ketabchi, S., Moosaei, H., Fallahi, S.: Optimal error correction of the absolute value equation using a genetic algorithm. Math. Comput. Modell. 57(9–10), 2339–2342 (2013)

    Google Scholar 

  22. Mangasarian, O.: A Newton method for linear programming. J. Optim. Theory Appl. 121(1), 1–18 (2004)

    MathSciNet  MATH  Google Scholar 

  23. Mangasarian, O.: Absolute value programming. Comput. Optim. Appl. 36(1), 43–53 (2007)

    MathSciNet  MATH  Google Scholar 

  24. Mangasarian, O.: A generalized Newton method for absolute value equations. Optim. Lett. 3(1), 101–108 (2009)

    MathSciNet  MATH  Google Scholar 

  25. Mangasarian, O., Meyer, R.: Absolute value equations. Linear Algebra Appl. 419(2–3), 359–367 (2006)

    MathSciNet  MATH  Google Scholar 

  26. Mangasarian, O.L.: Linear complementarity as absolute value equation solution. Optim. Lett. 8(4), 1529–1534 (2014)

    MathSciNet  MATH  Google Scholar 

  27. Mangasarian, O.L.: A hybrid algorithm for solving the absolute value equation. Optim. Lett. 9(7), 1469–1474 (2015)

    MathSciNet  MATH  Google Scholar 

  28. Moosaei, H., Ketabchi, S., Jafari, H.: Minimum norm solution of the absolute value equations via simulated annealing algorithm. Afrika Matematika 26(7–8), 1221–1228 (2015)

    MathSciNet  MATH  Google Scholar 

  29. Moosaei, H., Ketabchi, S., Noor, M.A., Iqbal, J., Hooshyarbakhsh, V.: Some techniques for solving absolute value equations. Appl. Math. Comput. 268, 696–705 (2015)

    MathSciNet  MATH  Google Scholar 

  30. Moosaei, H., Ketabchi, S., Pardalos, P.M.: Tikhonov regularization for infeasible absolute value equations. Optimization 65(8), 1531–1537 (2016)

    MathSciNet  MATH  Google Scholar 

  31. Nocedal, J., Wright, S.: Numerical Optimization. Springer, New York (2006)

    MATH  Google Scholar 

  32. Prokopyev, O.: On equivalent reformulations for absolute value equations. Comput. Optim. Appl. 44(3), 363–372 (2009)

    MathSciNet  MATH  Google Scholar 

  33. Raayatpanah, M., Moosaei, H., Pardalos, P.: Absolute value equations with uncertain data. Optim. Lett. 1–12 (2019, In press)

  34. Rohn, J.: Proofs to ’solving interval linear systems’. Freiburger Intervall-Berichte 84/7. Albert-Ludwigs-Universität, Freiburg (1984)

    Google Scholar 

  35. Rohn, J.: Systems of linear interval equations. Linear Algebra Appl. 126, 39–78 (1989)

    MathSciNet  MATH  Google Scholar 

  36. Rohn, J.: A theorem of the alternatives for the equation \(Ax+ B|x|= b\). Linear Multilinear Algebra 52(6), 421–426 (2004)

    MathSciNet  MATH  Google Scholar 

  37. Rohn, J.: On unique solvability of the absolute value equation. Optim. Lett. 3(4), 603–606 (2009)

    MathSciNet  MATH  Google Scholar 

  38. Rohn, J., Hooshyarbakhsh, V., Farhadsefat, R.: An iterative method for solving absolute value equations and sufficient conditions for unique solvability. Optim. Lett. 8(1), 35–44 (2014)

    MathSciNet  MATH  Google Scholar 

  39. Salahi, M., Ketabchi, S.: Correcting an inconsistent set of linear inequalities by the generalized Newton method. Optim. Methods Softw. 25(3), 457–465 (2010)

    MathSciNet  MATH  Google Scholar 

  40. Sherali, H.D., Tuncbilek, C.H.: A global optimization algorithm for polynomial programming problems using a reformulation–linearization technique. J. Global Optim. 2(1), 101–112 (1992)

    MathSciNet  MATH  Google Scholar 

  41. Tikhonov, A.N., Arsenin, V.I.: Solutions of Ill-Posed Problems. Winston, Washington, DC (1977)

    MATH  Google Scholar 

  42. Wu, S.L., Guo, P.: On the unique solvability of the absolute value equation. J. Optim. Theory Appl. 169(2), 705–712 (2016)

    MathSciNet  MATH  Google Scholar 

  43. Zamani, M., Hladík, M.: Error bounds and a condition number for the absolute value equations (2020). Preprint arXiv: 1912.12904

Download references

Acknowledgements

The work of H. Moosaei and M. Hladík was supported by the Czech Science Foundation Grant P403-18-04735S.

Author information

Authors and Affiliations

  1. Department of Applied Mathematics, Charles University, Malostranské nám. 25, 118 00, Prague, Czech Republic

    Hossein Moosaei & Milan Hladík

  2. Department of Applied Mathematics, Faculty of Mathematical Sciences, University of Guilan, Rasht, Iran

    Saeed Ketabchi

  3. Department of Mathematics, Faculty of Science, University of Bojnord, Bojnord, Iran

    Hossein Moosaei

Authors
  1. Hossein Moosaei
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. Saeed Ketabchi
    View author publications

    You can also search for this author inPubMed Google Scholar

  3. Milan Hladík
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to Milan Hladík.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Moosaei, H., Ketabchi, S. & Hladík, M. Optimal correction of the absolute value equations. J Glob Optim 79, 645–667 (2021). https://doi.org/10.1007/s10898-020-00948-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10898-020-00948-2

Keywords