Topographical Differential Evolution Using Pre-calculated Differentials

  • M. M. Ali
  • A. Törn
Part of the Nonconvex Optimization and Its Applications book series (NOIA, volume 59)


We present an algorithm for finding the global minimum of multimodal functions. The proposed algorithm is based on differential evolution (DE). Its distinguishing features are that it implements pre-calculated differentials and that it suitably utilizes topographical information on the objective function in deciding local search. These features are implemented in a periodic fashion. The algorithm has been tested on easy, moderately difficult test problems as well as on the difficult Lennard-Jones (LJ) potential function. Computational results using problems of dimensions upto 24 are reported. A robust computational behavior of the algorithm is shown.


Global optimization differential evolution pre-calculated differential continuous variable topographs graph minima 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Ratschek, H. and Rokne, J.: New Computer Methods for Global Optimization, Ellis Horwood, Chichester, 1988.Google Scholar
  2. [2]
    Törn, A. and Žilinskas, A.: Global Optimization, Springer-Verlag, Berlin, 1989.Google Scholar
  3. [3]
    Horst, R. and Tuy, H.: Global Optimization (Deterministic Approaches), Springer-Verlag, Berlin, 1990.Google Scholar
  4. [4]
    Wood, G. R.: Multidimensional bisection and global optimization, Comput. Math. Appl. 21 (1991), 161–172.CrossRefMathSciNetzbMATHGoogle Scholar
  5. [5]
    Floudas, A. and Pardalos, M.: Recent Advances in Global Optimization, Princeton University Press, Princeton, NJ, 1992.Google Scholar
  6. [6]
    Goldberg, D.: Genetic Algorithms in Search, Optimization and Machine Learning, Addison-Wesley, Reading, MA, 1989.Google Scholar
  7. [7]
    Ali, M. M., Törn, A. and Viitanen, S.: A numerical comparison of some modified controlled random search algorithms, J. Global Optim. 11 (1997), 377–385.MathSciNetGoogle Scholar
  8. [8]
    Storn, R. and Price, K.: Differential evolution — a simple and efficient heuristic for global optimization over continuous spaces, J. Global Optim. 11 (1997), 341–359.CrossRefMathSciNetGoogle Scholar
  9. [9]
    Dekkers, A. and Aarts, E.: Global optimization and simulated annealing, Math. Programming 50 (1991), 367–393.CrossRefMathSciNetGoogle Scholar
  10. [10]
    Ali, M. M. and Storey, C.: Aspiration based simulated annealing algorithms, J. Global Optim. 11 (1997), 181–191.MathSciNetGoogle Scholar
  11. [11]
    Ali, M. M., Storey, C. and Törn A.: Applications of stochastic global optimization algorithms to practical problems, J. Optim. Theory Appl. 95 (1997), 545–563.CrossRefMathSciNetGoogle Scholar
  12. [12]
    Rinnoy Kan, A. H. G. and Timmer, G. T.: Stochastic global optimization methods; Part II: Multilevel methods, Math. Programming 39 (1987), 57–78.MathSciNetGoogle Scholar
  13. [13]
    Törn, A. and Viitanen, S.: Topographical global optimization. In: A. Floudas and M. Pardalos (eds), Recent Advances in Global Optimization, Princeton University Press, Princeton, NJ, 1992.Google Scholar
  14. [14]
    Ali, M. M. and Storey, C.: Topographical multilevel single linkage, J. Global Optim. 5 (1994), 349–358.CrossRefMathSciNetGoogle Scholar
  15. [15]
    Ali, M. M. and Törn, A.: Evolution based global optimization techniques and the controlled random search algorithm: Proposed modifications and numerical studies, submitted to the J. Global Optim.Google Scholar
  16. [16]
    Ali, M. M. and Törn, A.: Optimization of carbon and silicon cluster geometry for Tersoff potential using differential evolution, In: A. Floudas and M. Pardalos (eds), Optimization in Computational Chemistry and Molecular Biology: Local and Global Approaches, Kluwer Academic Publishers, Dordrecht, 2000.Google Scholar
  17. [17]
    Tezuka, S. and ĽEcuyer, P.: Efficient and portable combined Tausworthe random number generators, ACM Trans. Modelling and Computer Simulation 1 (1991). 99–112.Google Scholar
  18. [18]
    Lui, D. C. and Nocedal, J.: On the limited memory BFGS method for large scale optimization, Math. Programming 45 (1989), 503–528.MathSciNetGoogle Scholar
  19. [19]
    Leary, R. H.: Global optima of Lennard-Jones clusters, J. Global Optim. 11 (1997), 35–53.CrossRefMathSciNetzbMATHGoogle Scholar

Copyright information

© Kluwer Academic Publishers 2002

Authors and Affiliations

  • M. M. Ali
    • 1
  • A. Törn
    • 2
  1. 1.Centre for Control Theory and Optimization Department of Computational and Applied MathematicsWitwatersrand UniversityJohannesburgSouth Africa
  2. 2.Department of Computer ScienceÅbo Akademi UniversityTurkuFinland

Personalised recommendations