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Journal of Global Optimization

, Volume 31, Issue 4, pp 635–672 | Cite as

A Numerical Evaluation of Several Stochastic Algorithms on Selected Continuous Global Optimization Test Problems

  • M. Montaz AliEmail author
  • Charoenchai Khompatraporn
  • Zelda B. Zabinsky
Article

Abstract

There is a need for a methodology to fairly compare and present evaluation study results of stochastic global optimization algorithms. This need raises two important questions of (i) an appropriate set of benchmark test problems that the algorithms may be tested upon and (ii) a methodology to compactly and completely present the results. To address the first question, we compiled a collection of test problems, some are better known than others. Although the compilation is not exhaustive, it provides an easily accessible collection of standard test problems for continuous global optimization. Five different stochastic global optimization algorithms have been tested on these problems and a performance profile plot based on the improvement of objective function values is constructed to investigate the macroscopic behavior of the algorithms. The paper also investigates the microscopic behavior of the algorithms through quartile sequential plots, and contrasts the information gained from these two kinds of plots. The effect of the length of run is explored by using three maximum numbers of function evaluations and it is shown to significantly impact the behavior of the algorithms.

Keywords

Controlled random search Differential evolution Empirical comparison of algorithms Genetic algorithm Global optimization Hide-and-Seek Improving Hit-and-Run Performance profile and Test problems Population set based global optimization Simulated annealing 

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References

  1. Ali, M.M., Storey, C. 1994Modified controlled random search algorithmsInternational Journal of Computer Mathematics54229235Google Scholar
  2. Ali, M.M., Storey, C., Törn, A. 1997Application of some stochastic global optimization algorithms to practical problemsJournal of Optimization Theory and Applications95545563MathSciNetGoogle Scholar
  3. Ali, M.M., Törn, A. 2004Population set based global optimization algorithms: Some modifications and numerical studiesComputers and Operations Research3117031725MathSciNetGoogle Scholar
  4. Ali, M.M., Törn, A. 2002Topographical differential evolution Algorithm using pre-calculated differentialsZilinskas, A.Dzemyda, G.Saltenis, V. eds. Stochastic and Global OptimizationKluwer Academic PublishersDordrecht/Boston/London117Google Scholar
  5. Aluffi-Pentini, F., Parisi, V., Zirilli, F. 1985Global optimization and stochastic differential equationsJournal of Optimization Theory and Applications47116Google Scholar
  6. Benke, K.K., Skinner, D.R. 1991A direct search algorithm for global optimization of multivariate functionsThe Australian Computer Journal237385Google Scholar
  7. Bersini H., Dorigo M., Langerman S., Seront G., Gambardella L. (1996). Results of the first International Contest on Evolutionary Optimization (1st ICEO). In: Proceedings of the IEEE International Conference on Evolutionary Computation (ICEC). pp. 611–615 IEEE Press, New YorkGoogle Scholar
  8. Bohachevsky, M.E., Johnson, M.E., Stein, M.L. 1986Generalized simulated annealing for function optimizationTechnometrics28209217Google Scholar
  9. Breiman, L., Cutler, A. 1993A deterministic algorithm for global optimizationMathematical Programming58179199Google Scholar
  10. Dekkers, A., Aarts, E. 1991Global optimization and simulated annealingMathematical Programming50367393Google Scholar
  11. Dixon, L., Szegö, G. 1975Towards Global OptimizationNorth HollandNew YorkGoogle Scholar
  12. Dixon, L., Szegö, G. 1978Towards Global Optimization. Vol. 2North HollandNew YorkGoogle Scholar
  13. Dolan, D., Moré, J. 2002Benchmarking Optimization Software with Performance ProfilesMathematical Programming Series A91201213Google Scholar
  14. Floudas, C.A., Pardalos, P.M. 1990A Collection of Test Problems for Constrained Global Optimization Algorithms, Lecture Notes in Computer Science, Vol. 455Springer-VerlagBerlin/Heidelberg/New YorkGoogle Scholar
  15. Floudas, C.A., Pardalos, P.M., Adjiman, C.S., Esposito, W.R., Gümüs, Z.H., Harding, S.T., Klepeis, J.L., Meyer, C.A., Schweiger, C.A. 1999Handbook of Test Problems in Local and Global OptimizationKluwer Academic PublishersDordrecht/Boston/LondonGoogle Scholar
  16. GAMS World (2002). GLOBAL Library, WWW-document, http://www.gamsworld.org/global/globallib.htmGoogle Scholar
  17. Goldberg, D. 1989Genetic Algorithm in Search, Optimization and Machine LearningAddison-Wesley Publishing CompanyReadingGoogle Scholar
  18. Gould N.I.M, Orban D., Toint P.L. (2001). CUTEr, a constrained and unconstrained testing environment, revisited, WWW-document, http://cuter.rl.ac.uk/cuter-www/problems.htmlGoogle Scholar
  19. Griewank, A.O. 1981Generalized Descend for Global OptimizationJournal of Optimization Theory and Applications341139Google Scholar
  20. Himmelblau, D.M. 1972Applied Nonlinear ProgrammingMcGraw-HillNew YorkGoogle Scholar
  21. Jansson, C., Knüppel, O. 1995A Branch and Bound Algorithm for Bound Constrained Optimization Problems without DerivativesJournal of Global Optimization7297331Google Scholar
  22. Khompatraporn, C., Pintér, J.D., Zabinsky, Z.B. 2005Comparative Assessment of Algorithms and Software for Global OptimizationJournal of Global Optimization31613633Google Scholar
  23. Levy, A.V., Montalvo, A. 1985The tunneling algorithm for the global minimization of functionsSociety for Industrial and Applied Mathematics61529Google Scholar
  24. McCormick, G.P. 1982Applied Nonlinear Programming, Theory, Algorithms and ApplicationsJohn Wiley and SonsNew YorkGoogle Scholar
  25. Metropolis, N., Rosenbluth, A.W., Rosenbluth, M.N., Teller, A.H., Teller, E. 1953Equation of state Calculations by fast computing machinesJournal of Chemical Physics2110871092Google Scholar
  26. Michalewicz, Z. 1996Genetic Algorithms + Data Structures = Evolution ProgramsSpringer-VerlagBerlin/Heidelberg/New YorkGoogle Scholar
  27. Moré, J., Garbow, B., Hillstrom, K. 1981Testing Unconstrained Optimization SoftwareACM Transaction on Mathematical Software71741Google Scholar
  28. Muhlenbein H., Schomisch S., Born J. (1991). The parallel genetic algorithm as function optimizer, In: Belew R., Booker L. (eds). Proceedings of the Fourth International Conference on Genetic Algorithms, Morgan Kaufman pp. 271–278Google Scholar
  29. Neumaier A. (2003a). COCONUT benchmark, WWW-document, http://www.mat.univie.ac.at/∼neum/glopt/coconut/benchmark.htmlGoogle Scholar
  30. Neumaier A. (2003b). Global and Local Optimization, WWW-document, http://solon.cma.univie.ac.at/∼neum/glopt.htmlGoogle Scholar
  31. Price, W.L. 1977Global optimization by controlled random searchComputer Journal20367370Google Scholar
  32. Price, W.L. 1983Global optimization by controlled random searchJournal of Optimization Theory and Applications40333348Google Scholar
  33. Price, W.L. 1987Global optimization algorithms for a CAD workstationJournal of Optimization Theory and Applications55133146MathSciNetGoogle Scholar
  34. Price K.V. (2002). Private Communication, 836 Owl Circle. Vacaville, CA 95687Google Scholar
  35. Romeijn, H.E., Smith, R.L. 1994Simulated annealing for constrained global optimizationJournal of Global Optimization5101126Google Scholar
  36. Salomon, R. 1995Reevaluating genetic algorithms performance under coordinate rotation of benchmark functionsBioSystems39263278Google Scholar
  37. Schwefel, H.P. 1995Evolution and Optimum SeekingJohn Wiley and SonsNew YorkGoogle Scholar
  38. Second (2nd) ICEO: Second International Contest on Evolutionary Optimization, WWW-document, http://iridia.ulb.ac.be/langerman/2ndICEO.htmlGoogle Scholar
  39. Smith, R.L. 1984Efficient Monte Carlo procedures for generating random feasible points uniformly over bounded regionsOperations Research3212961308Google Scholar
  40. Storn, R., Price, K. 1997Differential evolution: A simple and efficient heuristic for global optimization over continuous spacesJournal of Global Optimization11341359Google Scholar
  41. Törn, A., Žilinskas, A. 1989Global Optimization, Lecture Notes in Computer Science, Vol. 350Springer-VerlagBerlin/Heidelberg/New YorkGoogle Scholar
  42. Wolfe, M.A. 1978Numerical Methods for Unconstrained OptimizationVan Nostrand Reinhold CompanyNew YorkGoogle Scholar
  43. Wood, G.R., Zabinsky, Z.B. 2002Stochastic adaptive searchPardalos, P.Romeijn, E. eds. Handbook of Global OptimizationKluwer Academic PublishersDordrecht/Boston/London231249Google Scholar
  44. Zabinsky Z.B., Graesser D.L., Tuttle M.E., Kim G.I. (1992). Global optimization of composite laminates using improving hit and run. In: Floudas C., Pardalos P. (eds). Recent Advances in Global Optimization. Princeton University Press, pp. 343–368Google Scholar
  45. Zabinsky, Z.B., Smith, R.L., McDonald, J.F., Romeijn, H.E., Kaufman, D.E. 1993Improving hit-and-run for global optimizationJournal of Global Optimization3171192Google Scholar
  46. Zabinsky, Z.B., Wood, G.R. 2002Implementation of stochastic adaptive search with hit-and-run for a generatorPardolos, P.Romeijn, E. eds. Handbook of Global OptimizationKluwer Academic PublishersDordrecht/Boston/London251273Google Scholar

Copyright information

© Springer 2005

Authors and Affiliations

  • M. Montaz Ali
    • 1
    Email author
  • Charoenchai Khompatraporn
    • 2
  • Zelda B. Zabinsky
    • 2
  1. 1.School of Computational and Applied MathematicsWitwatersrand UniversityJohannesburgSouth Africa
  2. 2.Industrial EngineeringUniversity of WashingtonSeattleUSA

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