Skip to main content
Springer Nature Link
Log in

A new global optimization technique by auxiliary function method in a directional search

  • Original Paper
  • Published:
Optimization Letters Aims and scope Submit manuscript

Abstract

In this study, we introduce a new global optimization technique for a multi-dimensional unconstrained optimization problem. First, we present a new smoothing auxiliary function. Second, we transform the multi-dimensional problem into a one-dimensional problem by using an auxiliary function to reduce the number of local minimizers and then we find the global minimizer of the one-dimensional problem. Finally, we find the global minimizer of the multi-dimensional smooth objective function with the help of a new algorithm.

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

Similar content being viewed by others

References

  1. Robertson, B.L.: Direct search methods for nonsmooth problems using global optimization techniques. PhD thesis, University of Canterbury, Christchurch, New Zealand (2010)

  2. Bagirov, A.M., Rubinov, A.M., Zhang, J.: A multidimensional descent method for global optimization. Optimization 58, 611–625 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  3. Wu, Z.Y., Rubinov, A.M.: Global optimality conditions for some classes of optimization problems. J. Optim. Theory App. 145, 164–185 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  4. Wu, Z.Y., Li, G.Q., Quan, J.: Global optimality conditions and optimization methods for quadratic integer programming problems. J. Glob. Optim. 51, 549–568 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  5. Lin, Y., Yang, Y.: A new filled function method for constrained nonlinear equations. Appl. Math. Comput. 219, 3100–3112 (2012)

    MathSciNet  MATH  Google Scholar 

  6. Ling, B.W.K., Wu, C.Z., Teo, K.L., Rehbock, V.: Global optimal design of IIR filters via constraint transcription and filled function methods. Circ. Syst. Signal Process. 32, 1313–1334 (2013)

    Article  MathSciNet  Google Scholar 

  7. Locatelli, M., Maischberger, M., Schoen, F.: Differential evolution methods based on local searches. Comput. Oper. Res. 43, 169–180 (2014)

    Article  MATH  Google Scholar 

  8. Sahiner, A., Yilmaz, N., Kapusuz, G.: A descent global optimization method based on smoothing techniques via Bezier curves. Carpath. J. Math. 33, 373–380 (2017)

    MathSciNet  MATH  Google Scholar 

  9. Sahiner, A., Yilmaz, N., Demirozer, O.: Mathematical modeling and an application of the filled function method in entomology. Int. J. Pest Manag. 60, 232–237 (2014)

    Article  Google Scholar 

  10. Kearfott, R.B.: Interval extensions of non-smooth functions for global optimization and nonlinear systems solvers. Computation 57, 149–162 (1996)

    MathSciNet  MATH  Google Scholar 

  11. Wang, Y., Fan, L.: A smoothing evolutionary algorithm with circle search for global optimization. In: 4th IEEE International Conference, pp. 412–418 (2010)

  12. Fuduli, A., Gaudioso, M., Nurminski, E.A.: A splitting bundle approach for non-smooth non-convex minimization. Optimization 64, 1131–1151 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  13. Xu, Y., Zhang, Y., Wang, S.: A modified tunneling function method for non-smooth global optimization and its application in artificial neural network. Appl. Math. Model. 39, 6438–6450 (2015)

    Article  MathSciNet  Google Scholar 

  14. Zhang, Y., Zhang, L., Xu, Y.: New filled functions for nonsmooth global optimization. Appl. Math. Model. 33, 3114–3129 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  15. Bertsekas, D.P.: Nondifferentiable optimization via approximation. Math. Program. Stud. 3, 1–25 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  16. Zang, I.: A smoothing-out technique for min max optimization. Math. Program. 19, 61–77 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  17. Xavier, A.E.: Hyperbolic penalty: a new method for nonlinear programming with inequalities. Int. Trans. Oper. Res. 8, 659–671 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  18. Xavier, A.E., De Oliveira, A.A.F.: Optimal covering of plane domains by circles via hyperbolic smoothing. J. Glob. Optim. 31, 493–504 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  19. Griva, I., Nash, S.G., Sofer, A.: Linear and Nonlinear Optimization. SIM, Philadelphia (2009)

    Book  MATH  Google Scholar 

  20. Renpu, R.G.: A filled function method for finding a global minimizer of a function of several variables. Math. Program. 46, 191–204 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  21. Chen, X.: Smoothing methods for nonsmooth, nonconvex minimization. Math. Program. 134, 71–99 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  22. Yang, Y., Pang, L., Ma, X., Shen, J.: Constrained nonconvex nonsmooth optimization via proximal bundle method. J. Optim. Theory App. 163, 900–925 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  23. Ralph, D., Xu, H.: Implicit smoothing and its application to optimization with piecewise smooth equality constraints. J. Optim. Theory Appl. 124, 673–699 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  24. Wu, H., Zhang, P., Lin, G.H.: Smoothing approximations for some piecewise smooth functions. J. Oper. Res. Soc. China 3, 317–329 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  25. Yilmaz, N., Sahiner, A.: A new smoothing approximation to piecewise smooth functions and applications. Int. Conf. Anal. Appl. 226–226 (2016)

  26. Ma, S., Yang, Y., Liu, H.: A parameter free filled function for unconstrained global optimization. Appl. Math. Comput. 215, 3610–3619 (2010)

    MathSciNet  MATH  Google Scholar 

  27. Hedar, A.: Test functions for unconstrained global optimization. http://www-optima.amp.i.kyoto-u.ac.jp/member/student/hedar/Hedar-files/TestGO-files/Page364.htm. Accessed 2013

  28. Wei, F., Wang, Y., Lin, H.: A new filled function method with two parameters for global optimization. J. Optim. Theory Appl. 163, 510–527 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  29. Ketfi-Cherif, A., Ziadi, A.: Global descent method for constrained continuous global optimization. Appl. Math. Comput. 244, 209–221 (2014)

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Suleyman Demirel University, Isparta, Turkey

    Ahmet Sahiner

  2. Department of Mathematics, Suleyman Demirel University, Isparta, Turkey

    Shehab A. Ibrahem

Authors
  1. Ahmet Sahiner
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. Shehab A. Ibrahem
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to Ahmet Sahiner.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Sahiner, A., Ibrahem, S.A. A new global optimization technique by auxiliary function method in a directional search. Optim Lett 13, 309–323 (2019). https://doi.org/10.1007/s11590-018-1315-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11590-018-1315-1

Keywords