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On a smoothed penalty-based algorithm for global optimization

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Abstract

This paper presents a coercive smoothed penalty framework for nonsmooth and nonconvex constrained global optimization problems. The properties of the smoothed penalty function are derived. Convergence to an \(\varepsilon \)-global minimizer is proved. At each iteration k, the framework requires the \(\varepsilon ^{(k)}\)-global minimizer of a subproblem, where \(\varepsilon ^{(k)} \rightarrow \varepsilon \). We show that the subproblem may be solved by well-known stochastic metaheuristics, as well as by the artificial fish swarm (AFS) algorithm. In the limit, the AFS algorithm convergence to an \(\varepsilon ^{(k)}\)-global minimum of the real-valued smoothed penalty function is guaranteed with probability one, using the limiting behavior of Markov chains. In this context, we show that the transition probability of the Markov chain produced by the AFS algorithm, when generating a population where the best fitness is in the \(\varepsilon ^{(k)}\)-neighborhood of the global minimum, is one when this property holds in the current population, and is strictly bounded from zero when the property does not hold. Preliminary numerical experiments show that the presented penalty algorithm based on the coercive smoothed penalty gives very competitive results when compared with other penalty-based methods.

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References

  1. Ali, M.M., Golalikhani, M., Zhuang, J.: A computational study on different penalty approaches for solving constrained global optimization problems with the electromagnetism-like method. Optimization 63(3), 403–419 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  2. Ali, M.M., Zhu, W.X.: A penalty function-based differential evolution algorithm for constrained global optimization. Comput. Optim. Appl. 54(3), 707–739 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  3. Barbosa, H.J.C., Lemonge, A.C.C.: An adaptive penalty method for genetic algorithms in constrained optimization problems. In: Iba, H. (ed.) Frontiers in Evolutionary Robotics, pp. 9–34. I-Tech Education Publishing, Vienna (2008)

    Google Scholar 

  4. Coello, C.A.C.: Theoretical and numerical constraint-handling techniques used with evolutionary algorithms: a survey of the state of the art. Comput. Method. Appl. Mech. Eng. 191(11–12), 1245–1287 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  5. Liu, J.-L., Lin, J.-H.: Evolutionary computation of unconstrained and constrained problems using a novel momentum-type particle swarm optimization. Eng. Optim. 39(3), 287–305 (2007)

    Article  MathSciNet  Google Scholar 

  6. Petalas, Y.G., Parsopoulos, K.E., Vrahatis, M.N.: Memetic particle swarm optimization. Ann. Oper. Res. 156(1), 99–127 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  7. Silva, E.K., Barbosa, H.J.C., Lemonge, A.C.C.: An adaptive constraint handling technique for differential evolution with dynamic use of variants in engineering optimization. Optim. Eng. 12(1–2), 31–54 (2011)

    MATH  MathSciNet  Google Scholar 

  8. Bertsekas, D.P.: Nonlinear Programming, 2nd edn. Athena Scientific, Belmont (1999)

    MATH  Google Scholar 

  9. Mezura-Montes, E., Coello, C.A.C.: Constraint-handling in nature-inspired numerical optimization: past, present and future. Swarm Evol. Comput. 1(4), 173–194 (2011)

    Article  Google Scholar 

  10. Xavier, A.E.: Hyperbolic penalty: a new method for nonlinear programming with inequalities. Int. Trans. Oper. Res. 8(6), 659–671 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  11. Gonzaga, C.C., Castillo, R.A.: A nonlinear programming algorithm based on non-coercive penalty functions. Math. Program. Ser. A 96(1), 87–101 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  12. Auslender, A., Cominetti, R., Haddou, M.: Asymptotic analysis for penalty and barrier methods in convex and linear programming. Math. Oper. Res. 22(1), 43–62 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  13. Di Pillo, G., Lucidi, S., Rinaldi, F.: An approach to constrained global optimization based on exact penalty functions. J. Glob. Optim. 54(2), 251–260 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  14. Rocha, A.M.A.C., Fernandes, E.M.G.P., Martins, T.F.M.C.: Novel fish swarm heuristics for bound constrained global optimzation problems. In: Murgante, B. et al. (eds.) Computational Science and Its Applications, vol. 6784, Part III, pp. 185–199. ICCSA 2011, LNCS (2011)

  15. Boussaïd, I., Lepagnot, J., Siarry, P.: A survey on optimization metaheuristics. Inf. Sci. 237, 82–117 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  16. Gendreau, M., Potvin, J.-Y. (eds.): Handbook of Metaheuristics, 2nd edn. International Series in Operations Research and Management Science, Springer, Berlin (2010)

  17. Sörensen, K.: Metaheuristics—the metaphor exposed. Int. Trans. Oper. Res. 22, 3–18 (2015)

    Article  MATH  MathSciNet  Google Scholar 

  18. Griffin, J.D., Kolda, T.G.: Nonlinearly constrained optimization using heuristic penalty methods and asynchronous parallel generating set search. Appl. Math. Res. Express 2010(1), 36–62 (2010)

    MATH  MathSciNet  Google Scholar 

  19. Birgin, E.G., Floudas, C.A., Martínez, J.M.: Global minimization using an Augmented Lagrangian method with variable lower-level constraints. Math. Program. Ser. A 125(1), 139–162 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  20. Holland, J.H.: Adaptation in Natural and Artificial Systems. University of Michigan Press, Ann Arbor (1975)

    Google Scholar 

  21. Brest, J., Greiner, S., Bošković, B., Mernik, M., Žumer, V.: Self-adapting control parameters in differential evolution: a comparative study on numerical benchmark problems. IEEE Trans. Evol. Comput. 10, 646–657 (2006)

    Article  Google Scholar 

  22. Hansen, N., Ostermeier, A.: Completely derandomized self-adaptation in evolution strategies. Evol. Comput. 9(2), 159–195 (2001)

    Article  Google Scholar 

  23. Hansen, N.: The CMA evolution strategy: a comparing review. In: Lozano, J.A., Larranaga, P., Inza, I., Bengoetxea, E. (eds.) Towards a New Evolutionary Computation. Advances on Estimation of Distribution Algorithms, pp. 75–102. Springer, Berlin (2006)

    Chapter  Google Scholar 

  24. Socha, K., Dorigo, M.: Ant colony optimization for continuous domains. Eur. J. Oper. Res. 185(3), 1155–1173 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  25. Kirkpatrick, S., Gelatt, C., Vecchi, M.: Optimization by simulated annealing. Science 220, 671–680 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  26. Jiang, M., Wang, Y., Pfletschinger, S., Lagunas, M.A., Yuan, D.: Optimal multiuser detection with artificial fish swarm algorithm. In: Huang, D.-S., Heutte, L., Loog, M. (eds.) CCIS 2, ICIC 2007, pp. 1084–1093. Springer, Berlin (2007)

  27. Neshat, M., Sepidnam, G., Sargolzaei, M., Toosi, A.N.: Artificial fish swarm algorithm: a survey of the state-of-the-art, hybridization, combinatorial and indicative applications. Artif. Intell. Rev. 42(4), 965–997 (2014)

    Article  Google Scholar 

  28. Rocha, A.M.A.C., Martins, T.F.M.C., Fernandes, E.M.G.P.: An augmented Lagrangian fish swarm based method for global optimization. J. Comput. Appl. Math. 235(16), 4611–4620 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  29. Costa, M.F.P., Rocha, A.M.A.C., Fernandes, E.M.G.P.: An artificial fish swarm algorithm based hyperbolic augmented Lagrangian method. J. Comput. Appl. Math. 259(Part B), 868–876 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  30. Karr, A.F.: Probability, Springer Texts in Statistics. Springer, Berlin (1993)

    Google Scholar 

  31. Rudolph, G.: Convergence of evolutionary algorithms in general search spaces. In: Proceedings of Third IEEE Conference on Evolutionary Computation, pp. 50–54. IEEE Press, NJ (1996)

  32. Liang, J.J., Runarsson, T.P., Mezura-Montes, E., Clerc, M., Suganthan, P.N., Coello Coello, C.A., Deb, C.: Problem Definition and Evolution Criteria for the CEC 2006. In: Special Session on Constrained Real-Parameter Optimization, IEEE Congress on Evolutionary Computation, pp. 17–21. Vancouver, Canada, July (2006)

  33. Derrac, J., García, S., Molina, D., Herrera, F.: A practical tutorial on the use of nonparametric statistical tests as a methodology for comparing evolutionary and swarm intelligence algorithms. Swarm Evolut. Comput. 1(1), 3–18 (2011)

    Article  Google Scholar 

  34. Dolan, E.D., Moré, J.J.: Benchmarking optimization software with performance profiles. Math. Program. Ser. A 91(2), 201–213 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  35. Deb, K., Srivastava, S.: A genetic algorithm based augmented Lagrangian method for constrained optimization. Comput. Optim. Appl. 53(3), 869–902 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  36. Rocha, A.M.A.C., Costa, M.F.P., Fernandes, E.M.G.P.: A shifted hyperbolic augmented Lagrangian-based artificial fish two-swarm algorithm with guaranteed convergence for constrained global optimization. Eng. Optim. 48(12), 2114–2140 (2016)

    Article  MathSciNet  Google Scholar 

  37. Hooke, R., Jeeves, T.A.: Direct search solution of numerical and statistical problems. J. Assoc. Comput. Mach. 8(2), 212–229 (1961)

    Article  MATH  Google Scholar 

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Acknowledgements

The authors would like to thank two anonymous referees for their valuable comments and suggestions to improve the paper. This work has been supported by COMPETE: POCI-01-0145-FEDER-007043 and FCT—Fundação para a Ciência e Tecnologia within the projects UID/CEC/00319/2013 and UID/MAT/00013/2013.

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Authors and Affiliations

  1. Department of Production and Systems, Algoritmi Research Centre, University of Minho, Campus de Gualtar, 4710-057, Braga, Portugal

    Ana Maria A. C. Rocha

  2. Department of Mathematics and Applications, Centre of Mathematics, University of Minho, Campus de Gualtar, 4710-057, Braga, Portugal

    M. Fernanda P. Costa

  3. Algoritmi Research Centre, University of Minho, Campus de Gualtar, 4710-057, Braga, Portugal

    Edite M. G. P. Fernandes

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  1. Ana Maria A. C. Rocha
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Correspondence to Ana Maria A. C. Rocha.

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Rocha, A.M.A.C., Costa, M.F.P. & Fernandes, E.M.G.P. On a smoothed penalty-based algorithm for global optimization. J Glob Optim 69, 561–585 (2017). https://doi.org/10.1007/s10898-017-0504-2

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