Journal of Global Optimization

, Volume 14, Issue 4, pp 415–436 | Cite as

FRACTOP: A Geometric Partitioning Metaheuristic for Global Optimization

  • Melek Demirhan
  • Linet Özdamar
  • Levent Helvacıoğlu
  • Şevket Ilker Birbil


We propose a new metaheuristic, FRACTOP, for global optimization. FRACTOP is based on the geometric partitioning of the feasible region so that search metaheuristics such as Simulated Annealing (SA), or Genetic Algorithms (GA) which are activated in smaller subregions, have increased reliability in locating the global optimum. FRACTOP is able to incorporate any search heuristic devised for global optimization. The main contribution of FRACTOP is that it provides an intelligent guidance (through fuzzy measures) in locating the subregion containing the global optimum solution for the search heuristics imbedded in it. By executing the search in nonoverlapping subregions, FRACTOP eliminates the repetitive visits of the search heuristics to the same local area and furthermore, it becomes amenable for parallel processing. As FRACTOP conducts the search deeper into smaller subregions, many unpromising subregions are discarded from the feasible region. Thus, the initial feasible region gains a fractal structure with many space gaps which economizes on computation time. Computational experiments with FRACTOP indicate that the metaheuristic improves significantly the results obtained by random search (RS), SA and GA.

FRACTOP Geometric partitioning Fuzzy measures 


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  1. Androulakis, G.S. and Vrahatis, M.N. (1996), OPTAC: a portable software package for analyzing and comparing optimization methods by visualization, Journal of Computational and Applied Mathematics 72: 41-62.Google Scholar
  2. Armijo, L. (1966), Minimization of functions having Lipschitz continuous first partial derivatives, Pacific J. of Mathematics 16: 1-3.Google Scholar
  3. Beveridge, G. and Schechter, R. (1970), Optimization: Theory and Practice, Mc-Graw Hill, New York.Google Scholar
  4. Botsaris, C.A. (1978), A curvilinear optimization method based upon iterative estimation of the eigensystem of the hessian matrix, J. of Mathematical Analysis and Applications 63: 396-411.Google Scholar
  5. Bozyel, M.A. and Özdamar, L. (1997), A heuristic approach to the capacitated lot sizing and scheduling problem (with parallel facilities), Working Paper, Istanbul Kültür University, Dept. of Computer Engineering, Istanbul.Google Scholar
  6. Caprani, O., Gothaab, B. and Madsen, K. (1993), Use of real-valued local minimum in parallel interval global optimization, Interval Computations, 3: 71-82.Google Scholar
  7. Csendes, T. and Pinter, J. (1993), The impact of accelerating tools on the interval subdivision algorithm for global optimization, European Journal of Operations Research 65, 314-320.Google Scholar
  8. Davis, L. (ed.) (1987), Genetic Algorithms and Simulated Annealing, London, Pitman.Google Scholar
  9. Fletcher, R. and Powell, M. (1963), A rapidly convergent descent method for minimization, Computer Journal 6: 163-168.Google Scholar
  10. Fletcher, R. and Reeves, C. (1964), Function minimization by conjugate gradients, Computer Journal 7: 149-154.Google Scholar
  11. Gill, P. and Murray, W. (1972), Quasi-Newton methods for unconstrained optimization, J. Institute of Mathematics and its Applications, 9: 91-108.Google Scholar
  12. Glover, F. (1989), Tabu search-Part I, ORSA Journal of Computing 1: 190-206.Google Scholar
  13. Goldberg, D. (1989), Genetic Algorithms in Search, Optimization and Machine Learning, Addison-Wesley.Google Scholar
  14. Kan, A.H.G. and Timmer, G.T. (1984), Stochastic methods for global optimization, American J. of Mathematical Management Science 4: 7-40.Google Scholar
  15. Kearfott, B., An efficient degree-computation method for a generalized method of bisection, Numerische Mathematics 32: 109-127.Google Scholar
  16. Kirkpatrick, A., Gelatt, Jr., C.D. and Vechi, M.P. (1983), Optimization by simulated annealing, Science 220: 671-680.Google Scholar
  17. Klir, G. and Folger, T. (1988), Fuzzy Sets Uncertainty and Information, Prentice Hall, Englewood Cliffs, N.J.Google Scholar
  18. Michalewicz, Z. (1994), Genetic Algorithms + Data Structures = Evolution Programs, Springer Verlag, Berlin.Google Scholar
  19. More, B.J., Garbow, B.S. and Hillstrom, K.E. (1981), Testing unconstrained optimization, ACM Trans. Math. Software 7: 17-41.Google Scholar
  20. Özdamar, L. (1998), A genetic algorithm approach for the multi-mode resource-constrained project scheduling problem under general resource categories, to appear in IEEE Trans. On Systems, Man and Cybernetics.Google Scholar
  21. Özdamar, L. and Birbil, ¸ S.I. (1998), A hybrid genetic algorithm for the capacitated lot sizing and loading problem, to appear in European Journal of Operations Research.Google Scholar
  22. Özdamar, L. and Bozyel, M.A. (1998), Simultaneous lot sizing and loading of product families on parallel facilities of different classes, Int. J. Production Research, 36, 1305-1324.Google Scholar
  23. Pal, N.R. and Bezdek, J.C. (1994), Measuring fuzzy uncertainty, IEEE Transactions on Fuzzy Systems 2, 107-118.Google Scholar
  24. Pal, N.R., Bezdek, J.C. and Hemasinha, R. (1992), Uncertainty measures for evidential reasoning I: A review, Int. J. of Approximate Reasoning 7: 165-183.Google Scholar
  25. Pal, N.R., Bezdek, J.C. and Hemasinha, R. (1993), Uncertainty measures for evidential reasoning II: A review Int. J. of Approximate Reasoning, 8, 1-16.Google Scholar
  26. Press, W.H., Teukolsky, S.A., Vetterling, W.T. and Flannery, B.P. (1992), Numerical Recipes, The Art of Scientific Computing. Cambridge University Press, New York.Google Scholar
  27. Ross, T.J. (1995), Fuzzy Logic with Engineering Applications, Mc-Graw Hill.Google Scholar
  28. Schaffer, J.D. (1989), A study of control parameters affecting online performance of genetic algorithms for function optimization, Proc. of the Third Int. Con. on Genetic Algorithms 51-60.Google Scholar
  29. Srinivas, M. and Patnaik, L.M. (1994), Adaptive probabilities of crossover and mutation in genetic algorithms, IEEE Transactions on Systems, Man and Cybernetics 24: 656-667.Google Scholar
  30. Stenger, F. (1975), Computing the topological degree of a mapping in Rn, Numerische Mathematics 25: 23-38.Google Scholar
  31. Tempelmeier, H. and Derstroff, M. (1996), A Lagrangean based heuristic for dynamic multi item multi level constrained lot sizing with setup times, Management Science 42: 738-757.Google Scholar
  32. Törn, A. and Viitanen, S. (1994), Topographical global optimization using pre-sampled points, Journal of Global Optimization 5: 267-276.Google Scholar
  33. Vrahatis, M.N. (1988), Solving systems of nonlinear equations using the nonzero value of the topological degree, ACM Trans. Math. Software 14: 312-329.Google Scholar
  34. Yager, R. (1983), Entropy and specificity in a mathematical theory of evidence, Int. J. of General Systems 9: 149-260.Google Scholar
  35. Zimmermann, H. (1991), Fuzzy Set Theory and Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands.Google Scholar

Copyright information

© Kluwer Academic Publishers 1999

Authors and Affiliations

  • Melek Demirhan
    • 1
  • Linet Özdamar
    • 2
  • Levent Helvacıoğlu
    • 2
  • Şevket Ilker Birbil
    • 1
  1. 1.Department of Systems EngineeringYeditepe UniversityTurkey
  2. 2.Department of Computer EngineeringIstanbul Kültür UniversityIstanbulTurkey

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