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A Hybrid Simplex Search for Global Optimization with Representation Formula and Genetic Algorithm

  • Hafid ZidaniEmail author
  • Rachid Ellaia
  • Eduardo Souza de Cursi
Conference paper
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 991)

Abstract

We consider the problem of minimizing a given function \(f:\mathbb {R}^n\longrightarrow \mathbb {R}\) on a regular not empty closed set S. When f attains a global minimum at exactly one point \(x^* \in S\), for a convenient random variable X and a convenient function \(g:\mathbb {R}^2\longrightarrow \mathbb {R}\). In this paper, we propose to use this Representation Formula (RF) to numerically generate an initial population. In order to obtain a more accurate results, the Representation Formula has been coupled with other algorithms:
  • Classical Genetic Algorithm (GA). We obtain a new algorithm called (RFGA),

  • Genetic Algorithm using Nelder Mead algorithm at the mutation stage (GANM). We obtain a new algorithm called (RFGANM),

  • Nelder Mead Algorithm. We obtain a new algorithm called (RFNM).

All these six algorithms (RF, GA, RFGA, GANM, RFGANM, RFNM) were tested on 21 benchmark functions with a complete analysis of the effect of different parameters of the methods. The experiments show that the RFNM is the most successful algorithm. Its performance was compared with the other algorithms, and observed to be the more effective, robust, and stable than the others.

Keywords

Global optimization Genetic algorithm Representation formula Nelder Mead algorithm 

References

  1. 1.
    Chelouah, R., Siarry, P.: Genetic and Nelder-Mead algorithms hybridized for a more accurate global optimization of continuous multiminima functions. Eur. J. Oper. Res. 148(2), 335–348 (2003)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Chelouah, R., Siarry, P.: A hybrid method combining continuous tabu search and nelder-mead simplex algorithms for the global optimiza tion of multiminima functions. Eur. J. Oper. Res. 161(3), 636–654 (2005)CrossRefGoogle Scholar
  3. 3.
    Souza de Cursi, J.: Representation of solutions in variational calculus. In: Variational Formulations in Mechanics: Theory and Applications, pp. 87–106 (2007)Google Scholar
  4. 4.
    Souza de Cursi, J., El Hami, A.: Representation of solutions in continuous optimization. Int. J. Simul. Multidiscip. Design Optim. (2009)Google Scholar
  5. 5.
    Floudas, C.A., Pardalos, P.M. (eds.).: Encyclopedia of Optimization. Springer, Boston (2009)Google Scholar
  6. 6.
    Gaviano, M., Kvasov, D., Lera, D., Sergeyev, Y.D.: Software for generation of classes of test functions with known local and global minima for global optimization. ACM Trans. Math. Softw. 29(4), 469–480 (2003)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Georgievaa, A., Jordanov, I.: A hybridmeta-heuristic for global optimisation using low-discrepancy sequences of points. Comput. Oper. Res. 37(3), 456–469 (2010)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Ivorra, B., Mohammadi, B., Ramos, A.M., Redont, I.: Optimizing initial guesses to improve global minimization. In: Pre-Publication of the Department of Applied Mathematics MA-UCM-UCM-No 2008-06 - Universidad Complutense de Madrid, 3 Plaza de Ciencias, 28040, Madrid, Spain. p. 17 (2008)Google Scholar
  9. 9.
    Pincus, M.: A closed formula solution of certain programming problems. Oper. Res. 16(3), 690–694 (1968)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  • Hafid Zidani
    • 1
    • 2
    Email author
  • Rachid Ellaia
    • 1
  • Eduardo Souza de Cursi
    • 2
  1. 1.LERMA, Mohammed V University - Engineering Mohammedia School, RabatAgdalMorocco
  2. 2.Laboratory of Mechanics of NormandyNational Institute for Applied Sciences - RouenSt Etienne du Rouvray CedexFrance

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