On initial populations of a genetic algorithm for continuous optimization problems

  • Heikki Maaranen
  • Kaisa Miettinen
  • Antti Penttinen
Original Paper

Abstract

Genetic algorithms are commonly used metaheuristics for global optimization, but there has been very little research done on the generation of their initial population. In this paper, we look for an answer to the question whether the initial population plays a role in the performance of genetic algorithms and if so, how it should be generated. We show with a simple example that initial populations may have an effect on the best objective function value found for several generations. Traditionally, initial populations are generated using pseudo random numbers, but there are many alternative ways. We study the properties of different point generators using four main criteria: the uniform coverage and the genetic diversity of the points as well as the speed and the usability of the generator. We use the point generators to generate initial populations for a genetic algorithm and study what effects the uniform coverage and the genetic diversity have on the convergence and on the final objective function values. For our tests, we have selected one pseudo and one quasi random sequence generator and two spatial point processes: simple sequential inhibition process and nonaligned systematic sampling. In numerical experiments, we solve a set of 52 continuous test functions from 16 different function families, and analyze and discuss the results.

Keywords

Global optimization Continuous variables Evolutionary algorithms Initial population Random number generation 

References

  1. 1.
    Acworth P., Broadie M., Glasserman P.(1998). A comparison of some Monte Carlo and quasi Monte Carlo techniques for option pricing. In: Niederreiter H., Hellekalek P., Larcher G., Zinterhof P. (eds). Monte Carlo and Quasi-Monte Carlo Methods 1996, number 127 in Lecture notes in statistics. Springer-Verlag, New York, pp 1–18Google Scholar
  2. 2.
    Ali M.M., Storey C.(1994). Modified controlled random search algorithms. Int. J. Comput. Mathemat. 53, 229–235CrossRefGoogle Scholar
  3. 3.
    Ali M.M., Storey C.(1994). Topographical multilevel single linkage. J. Global Optimizat. 5(4): 349–358CrossRefGoogle Scholar
  4. 4.
    Bratley P., Fox B.L.(1988). Algorithm 659: implementing Sobol’s quasirandom sequence generator. ACM Trans. Mathemat. Software 14(1): 88–100CrossRefGoogle Scholar
  5. 5.
    Bratley P., Fox B.L., Niederreiter H.(1992). Implementation and tests of low-discrepancy sequences. ACM Trans. Model. Comput. Simulat. 2(3): 195–213CrossRefGoogle Scholar
  6. 6.
    Couture R., L’Ecuyer P.(1997). Distribution properties of multiply-with-carry random number Mathemat. Computat. 66(218): 591–607CrossRefGoogle Scholar
  7. 7.
    Dekkers A., Aarts E.(1991). Global optimization and simulated annealing. Mathemat. Program. 50(3): 367–393CrossRefGoogle Scholar
  8. 8.
    DIEHARD: a battery of tests for random number generators developed by George Marsaglia. http://stat.fsu.edu~geo/diehard.html, September (2004).Google Scholar
  9. 9.
    Diggle P.J.(1983). Statistical Analysis of Spatial Point Patterns. Academic Press, LondonGoogle Scholar
  10. 10.
    Dykes, S., Rosen, B.: Parallel very fast simulated reannealing by temperature block partitioning. In Proceedings of the 1994 IEEE International Conference on Systems, Man, and Cybernetics, vol. 2, pp. 1914–1919 (1994)Google Scholar
  11. 11.
    Eiben A.E., Hinterding R., Michalewicz Z.(1999). Parameter control in evolutionary algorithms. IEEE Trans. Evol. Computat. 3(2): 124–141CrossRefGoogle Scholar
  12. 12.
    Floudas C.A., Pardalos P.M. (eds.)(1992). Recent Advances in Global Optimization. Princeton University Press, Princeton NJGoogle Scholar
  13. 13.
    Genetic and evolutionary algorithm toolbox for use with Matlab. http://www.geatbx.com/, June (2004)Google Scholar
  14. 14.
    Gentle J.E.(1998). Random Number Generation and Monte Carlo Methods. Springer-Verlag, New YorkGoogle Scholar
  15. 15.
    Glover F., Kochenberger G.A. (eds)(2003). Handbook of Metaheuristics. Kluwer Academic Publishers, BostonGoogle Scholar
  16. 16.
    Goldberg D.E.(1989). Genetic Algorithms in Search, Optimization, and Machine Learning. Wesley, New YorkGoogle Scholar
  17. 17.
    Halton J.H.(1960). On the efficiency of certain quasi-random sequences of points in evaluating dimentional integrals. Numerische Mathematik, 2, 84–90CrossRefGoogle Scholar
  18. 18.
    Horst R., Pardalos P.M. (eds)(1995). Handbook of Global Optimization. Kluwer Academic Publishers, DordrechtGoogle Scholar
  19. 19.
    Kingsley M.C.S., Kanneworff P., Carlsson D.M.(2004). Buffered random sampling: a sequential inhibited spatial point process applied to sampling in a trawl survey for northern shrimp pandalus borealis in west Greenland waters. ICES J. Mar. Sci. 61(1): 12–24CrossRefGoogle Scholar
  20. 20.
    Larcher G.(2004). Digital point sets: analysis and applications. In: Hellekalek P., Larcher G. (eds) Random and Quasi Random Point Sets, number 138 in Lecture Notes in Statistics. Springer-Verlag, New York, pp. 167–222Google Scholar
  21. 21.
    Lei G.(2002). Adaptive random search in quasi-Monte Carlo methods for global optimization. Comput. Mathemat. Appl. 43(6–7), 747–754Google Scholar
  22. 22.
    Lemieux C., L’Ecuyer P.(2001). On selection criteria for lattice rules and other quasi-Monte Carlo point sets. Mathemat. Comput. Simulat. 55(1–3): 139–148CrossRefGoogle Scholar
  23. 23.
    Maaranen, H.: On Global Optimization with Aspects to Method Comparison and Hybridization. Licentiate thesis, Department of Mathematical Information Technology, University of Jyväskylä (2002)Google Scholar
  24. 24.
    Maaranen H., Miettinen K., Mäkelä M.M.(2004). Quasi-random initial population for genetic algorithms. Comput. Mathemat. Appl. 47(12): 1885–1895CrossRefGoogle Scholar
  25. 25.
    Michalewicz Z.(1994). Genetic Algorithms + Data Structures = Evolution Programs. Springer-Verlag, BerlinGoogle Scholar
  26. 26.
    Michalewicz Z., Logan T.D., Swaminathan S.(1994). Evolutionary operators for continuous convex parameter spaces. In: Sebald A.V., Fogel L.J. (eds) Proceedings of the 3rd Annual Conference on Evolutionary Programming. World Scientific Publishing, River Edge NJ, pp. 84–97Google Scholar
  27. 27.
    Morokoff W.J., Caflisch R.E.(1994). Quasi-random sequences and their discrepancies. SIAM J. Sci. Comput. 15(6): 1251–1279CrossRefGoogle Scholar
  28. 28.
    Neter J., Wasserman W., Whitmore G.A.(1988). Applied Statistics. Allyn and Bacon, Inc., Boston, third editionGoogle Scholar
  29. 29.
    Niederreiter H.(1983). Quasi-Monte Carlo methods for global optimization. In: Grossmann W., Pflug G., Vincze I., Wertz W. (eds) Proceedings of the 4th Pannonian Symposium on Mathematical Statistics. pp. 251–267Google Scholar
  30. 30.
    Niederreiter H.(1992). Random Number Generation and Quasi-Monte Carlo Methods. SIAM, PhiladelphiaGoogle Scholar
  31. 31.
    Nix A.E., Vose M.D.(1992). Modeling genetic algorithms with Markov chains. Ann. Mathemat. Arti. Intell. 5(1): 79–88CrossRefGoogle Scholar
  32. 32.
    Pardalos P.M., Romeijn H.E. (eds)(2002). Handbook of Global Optimization, vol. 2. Kluwer Academic Publishers, BostonGoogle Scholar
  33. 33.
    Press W.H., Teukolsky S.A.(1989). Quasi- (that is, sub-) random numbers. Comput. Phys. 3(6): 76–79Google Scholar
  34. 34.
    Ripley B.D.(1981). Spatial Statistics. John Wiley & Sons, New YorkGoogle Scholar
  35. 35.
    Snyder W.C.(2000). Accuracy estimation for quasi-Monte Carlo simulations. Mathemat Comput. Simulat. 54(1–3): 131–143CrossRefGoogle Scholar
  36. 36.
    Sobol’ I.M.(1998). On quasi-Monte Carlo integrations. Mathemat. Comput. Simulat. 47(2–5): 103–112CrossRefGoogle Scholar
  37. 37.
    Sobol’ I.M., Bakin S.G.(1994). On the crude multidimensional search. J. Computat. Appl. Mathemat. 56(3): 283–293CrossRefGoogle Scholar
  38. 38.
    Sobol I.M., Tutunnikov A.V.(1996). A varience reducing multiplier for Monte Carlo integrations. Mathemat. Comput. Model. 23(8–9): 87–96CrossRefGoogle Scholar
  39. 39.
    Tang H.-C.(2003). Using an adaptive genetic algorithm with reversals to find good second-order multiple recursive random number generators. Mathemat. Methods Operat. Res. 57(1): 41–48CrossRefGoogle Scholar
  40. 40.
    Test problems for global optimization. http://www.imm.dtu.dk/~km/GlobOpt/testex/, June (2004)Google Scholar
  41. 41.
    Test problems in R 2. http://iridia.ulb.ac.be/~aroli/ICEO/Functions/Functions.html, June (2004)Google Scholar
  42. 42.
    Trafalis B., Kasap S.(2002). A novel metaheuristics approach for continuous global optimization. J. Global Optimizat. 23(2): 171–190CrossRefGoogle Scholar
  43. 43.
    Tuffin B.(1996). On the use of low discrepancy sequences in Monte Carlo methods. Monte Carlo Methods Appl. 2(4): 295–320CrossRefGoogle Scholar
  44. 44.
    Weisstein, E.W.: “circle packing”. From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/CirclePacking.html, September (2004)Google Scholar
  45. 45.
    Wikramaratna R.S.(1989). ACORN—a new method for generating sequences of uniformly distributed pseudo-random numbers. J. Computat. Phys. 83(1): 16–31CrossRefGoogle Scholar
  46. 46.
    Wright A.H., Zhao Y.(1999). Markov chain models of genetic algorithms. In: Banzhaf W., Daida J., Eiben A.E., Garzon M.H., Honavar V., Jakiela M., Smith R.E. (eds) GECCO-99: Proceedings of the Genetic and Evolutionary Computation Conference. Morgan Kauffman, SanMateo, California, pp. 734–741Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2006

Authors and Affiliations

  • Heikki Maaranen
    • 1
  • Kaisa Miettinen
    • 2
  • Antti Penttinen
    • 3
  1. 1.Patria Aviation OyHalliFinland
  2. 2.Helsinki School of EconomicsHelsinkiFinland
  3. 3.Department of Mathematics and StatisticsUniversity of JyväskyläyväskyläFinland

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