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Optimization Letters

, Volume 13, Issue 2, pp 309–323 | Cite as

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

  • Ahmet SahinerEmail author
  • Shehab A. Ibrahem
Original Paper
  • 91 Downloads

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.

Keywords

Global optimization Non-smooth optimization Smoothing technique Directional search 

References

  1. 1.
    Robertson, B.L.: Direct search methods for nonsmooth problems using global optimization techniques. PhD thesis, University of Canterbury, Christchurch, New Zealand (2010)Google Scholar
  2. 2.
    Bagirov, A.M., Rubinov, A.M., Zhang, J.: A multidimensional descent method for global optimization. Optimization 58, 611–625 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Wu, Z.Y., Rubinov, A.M.: Global optimality conditions for some classes of optimization problems. J. Optim. Theory App. 145, 164–185 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Lin, Y., Yang, Y.: A new filled function method for constrained nonlinear equations. Appl. Math. Comput. 219, 3100–3112 (2012)MathSciNetzbMATHGoogle Scholar
  6. 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)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Locatelli, M., Maischberger, M., Schoen, F.: Differential evolution methods based on local searches. Comput. Oper. Res. 43, 169–180 (2014)CrossRefzbMATHGoogle Scholar
  8. 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)MathSciNetzbMATHGoogle Scholar
  9. 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)CrossRefGoogle Scholar
  10. 10.
    Kearfott, R.B.: Interval extensions of non-smooth functions for global optimization and nonlinear systems solvers. Computation 57, 149–162 (1996)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Wang, Y., Fan, L.: A smoothing evolutionary algorithm with circle search for global optimization. In: 4th IEEE International Conference, pp. 412–418 (2010)Google Scholar
  12. 12.
    Fuduli, A., Gaudioso, M., Nurminski, E.A.: A splitting bundle approach for non-smooth non-convex minimization. Optimization 64, 1131–1151 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 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)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Zhang, Y., Zhang, L., Xu, Y.: New filled functions for nonsmooth global optimization. Appl. Math. Model. 33, 3114–3129 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Bertsekas, D.P.: Nondifferentiable optimization via approximation. Math. Program. Stud. 3, 1–25 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Zang, I.: A smoothing-out technique for min max optimization. Math. Program. 19, 61–77 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Xavier, A.E.: Hyperbolic penalty: a new method for nonlinear programming with inequalities. Int. Trans. Oper. Res. 8, 659–671 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Griva, I., Nash, S.G., Sofer, A.: Linear and Nonlinear Optimization. SIM, Philadelphia (2009)CrossRefzbMATHGoogle Scholar
  20. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Chen, X.: Smoothing methods for nonsmooth, nonconvex minimization. Math. Program. 134, 71–99 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Wu, H., Zhang, P., Lin, G.H.: Smoothing approximations for some piecewise smooth functions. J. Oper. Res. Soc. China 3, 317–329 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Yilmaz, N., Sahiner, A.: A new smoothing approximation to piecewise smooth functions and applications. Int. Conf. Anal. Appl. 226–226 (2016)Google Scholar
  26. 26.
    Ma, S., Yang, Y., Liu, H.: A parameter free filled function for unconstrained global optimization. Appl. Math. Comput. 215, 3610–3619 (2010)MathSciNetzbMATHGoogle Scholar
  27. 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. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Ketfi-Cherif, A., Ziadi, A.: Global descent method for constrained continuous global optimization. Appl. Math. Comput. 244, 209–221 (2014)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsSuleyman Demirel UniversityIspartaTurkey
  2. 2.Department of MathematicsSuleyman Demirel UniversityIspartaTurkey

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