Skip to main content
SpringerLink
Log in

Minimum norm solution of the absolute value equations via simulated annealing algorithm

  • Published:
Afrika Matematika Aims and scope Submit manuscript

Abstract

This paper mainly aims to obtain a minimum norm solution to the NP-hard absolute value equations (AVE) \(Ax-|x|=b\). In our proposed method, first we show that the AVE is equivalent to a bilinear programming problem and then we present a system tantamount to this problem. To find the minimum norm solution to the AVE, we solve a quadratic programming problem with quadratic and linear constraints. Eventually, we solve the obtained programming problem by Simulated Annealing algorithm. Numerical results support the efficiency of the proposed method and the high accuracy of calculation.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Pardalos, P.M., Ketabchi, S., Moosaei, H.: Minimum norm solution to the positive semidefinite linear complementarity problem. Optimization 63(3), 359–369 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  2. Pardalos, P.M., Rosen, J.B.: Global optimization approach to the linear complementarity problem. SIAM J. Sci. Stat. Comput. 9, 341–353 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  3. Pardalos, P.M., Ye, Y.: Class of linear complementarity problems solvable in polynomial time. Linear Algebra Appl. 152, 3–17.99 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  4. Bazaraa, M.S., Sherali, H.D., Shetty, C.M.: Nonlinear Programming Therory and Algorithms. Wiely, USA (1999)

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  6. Cottle, R.W., Dantzig, G.: The Linear Complementarity Problem. Academic Press, New York (1992)

    MATH  Google Scholar 

  7. Mangasarian, O.L., Meyer, R.R.: Absolute value equations. Linear Algebra Appl. 419, 359367 (2006)

    Article  MathSciNet  Google Scholar 

  8. Mangasarian, O.L.: A generalized Newton method for absolute value equations. Optim. Lett. 3(1), 101108 (2009)

    Article  MathSciNet  Google Scholar 

  9. 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)

    Article  MATH  MathSciNet  Google Scholar 

  10. Longquan, Y.: Particle swarm optimization for absolute value equations. J. Comput. Inf. Syst. 7, 2359–2366 (2010)

    Google Scholar 

  11. Noor, M.A., Iqbal, J., Noor, K.I., Al-Said, E.: On an iterative method for solving absolute value equations. Optim. Lett. doi: 10.1007/s11590-011-0332-0

  12. Noor, M.A., Iqbal, J., Khattri, S., Al-Said, E.: A new iterative method for solving absolute value equations. Int. J. Phys. Sci. 6(7), 1793–1797 (2011)

    Google Scholar 

  13. Mangasarian, O.L.: Absolute value programming. Comput. Optim. Appl. 36(1), 4353 (2007)

    Article  MathSciNet  Google Scholar 

  14. Mangasarian, O.L.: Absolute value equation solution via concave minimization. Optim. Lett. 1, 38 (2007)

    MathSciNet  Google Scholar 

  15. Yong, L., Liu, S., Zhang, S., Deng, F.: A new method for absolute value equations based on harmony search algorithm. ICIC Express Lett. Part B Appl. 2(6), 1231–1236 (2011)

  16. Rohn, J.: An algorithm for computing all solutions of an absolute value equation. Optim. Lett. 6(5), 851–856 (2012)

  17. Kirkpatrick, S., Gelatt, C.D., Vecchi, M.P.: Optimization by Simulated Annealing. 220, 671–680 (1983)

  18. Corny, A.: Theorem dynamical approach to the TSP. J. Optim. Theory Appl. 45, 41–51 (1985)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

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

    Hossein Moosaei

  2. Department of Applied Mathematics, Faculty of Mathematical Sciences, University of Guilan, P.O. Box 1914, Rasht, Iran

    Saeed Ketabchi

  3. Department of Industrial Engineering, Sharif University of Technology, Tehran, Iran

    Hamed Jafari

Authors
  1. Hossein Moosaei
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Saeed Ketabchi
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Hamed Jafari
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hossein Moosaei.

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. & Jafari, H. Minimum norm solution of the absolute value equations via simulated annealing algorithm. Afr. Mat. 26, 1221–1228 (2015). https://doi.org/10.1007/s13370-014-0281-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13370-014-0281-8

Keywords

Mathematics Subject Classfication

Search

Navigation