Journal of Global Optimization

, Volume 47, Issue 2, pp 293–325 | Cite as

The oracle penalty method

Article

Abstract

A new and universal penalty method is introduced in this contribution. It is especially intended to be applied in stochastic metaheuristics like genetic algorithms, particle swarm optimization or ant colony optimization. The novelty of this method is, that it is an advanced approach that only requires one parameter to be tuned. Moreover this parameter, named oracle, is easy and intuitive to handle. A pseudo-code implementation of the method is presented together with numerical results on a set of 60 constrained benchmark problems from the open literature. The results are compared with those obtained by common penalty methods, revealing the strength of the proposed approach. Further results on three real-world applications are briefly discussed and fortify the practical usefulness and capability of the method.

Keywords

Constrained optimization Global optimization Penalty function Stochastic metaheuristic Ant colony optimization MIDACO-Solver Mixed integer nonlinear programming (MINLP) 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Asaadi J.: A computational comparison of some non-linear programs. Math. Program. 4, 144–154 (1973)CrossRefGoogle Scholar
  2. 2.
    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. M. 191, 1245–1287 (2002)CrossRefGoogle Scholar
  3. 3.
    Coit, D.W., Smith, A.E.: Penalty Guided Genetic Search for Reliability Design Optimization. Comput. Ind. Eng. 30, Special issue on genetic algorithms, pp. 895–904 (1996)Google Scholar
  4. 4.
    Dahl H., Meeraus A., Zenios S.A.: Some financial optimization models: risk management. In: Zenios, S.A. (eds) Financial Optimization, Cambridge University Press, New York (1993)Google Scholar
  5. 5.
    Dorigo M., Stuetzle T.: Ant Colony Optimization. MIT Press, Cambridge (2004)Google Scholar
  6. 6.
    Downs J.J., Vogel E.F.: Plant-wide industrial process control problem. Comput. Chem. Eng. 17, 245–255 (1993)CrossRefGoogle Scholar
  7. 7.
    Duran M., Grossmann I.E.: An outer-approximation algorithm for a class of mixed-integer nonlinear programs. Math. Program. 36, 307–339 (1986)CrossRefGoogle Scholar
  8. 8.
    Egea J.A., Rodríguez-Fernández M., Banga J.R., Martí R.: Scatter search for chemical and bio-process optimization. J. Glob. Optim. 37, 481–503 (2007)CrossRefGoogle Scholar
  9. 9.
    Exler O., Schittkowksi K.: A trust region SQP algorithm for mixed-integer nonlinear programming. Optim. Lett. 3, 269–280 (2007)CrossRefGoogle Scholar
  10. 10.
    Exler O., Antelo L.T., Egea J.A., Alonso A.A., Banga J.R.: A Tabu search-based algorithm for mixed-integer nonlinear problems and its application to integrated process and control system design. Comput. Chem. Eng. 32, 1877–1891 (2008)CrossRefGoogle Scholar
  11. 11.
    Floudas C.A., Pardalos P.M., Adjiman C.S., Esposito W.R., Gumus Z.H., Harding S.T., Klepeis J.L., Meyer C.A., Stuetzle C.A.: Handbook of Test Problems in Local and Global Optimization. Kluwer Academic Publishers, Dordrecht (1999)Google Scholar
  12. 12.
    Floudas C.A.: Nonlinear and Mixed Integer Optimization: Fundamentals and Applications. Oxford University Press, Oxford (1995)Google Scholar
  13. 13.
    Floudas, C.A., Pardalos, P.M.: Collection of Test Problems for Constrained Global Optimization Algorithms. Lecture Notes in Computer Science 455, Springer, New York (1990)Google Scholar
  14. 14.
    Goldberg D.E.: Genetic Algorithms in Search, Optimization and Machine Learning. Kluwer Academic Publishers, Boston (1989)Google Scholar
  15. 15.
    Glover F., Laguna M., Marti R.: Fundamentals of scatter search and path relinking. Control Cybern. 39, 653–684 (2000)Google Scholar
  16. 16.
    Grossmann, I.E., Kravanja, Z.: Mixed-integer nonlinear programming: A survey of algorithms and applications. The IMA Volumes in Mathematics and its Applications, vol. 93, Large Scale Optimization with Applications. Part II: Optimal Design and Control, Biegler, T.F., Coleman, T.F., (eds.), pp. 73–100, Springer, New York (1997)Google Scholar
  17. 17.
    Gupta O.K., Ravindran V.: Branch and bound experiments in convex non-linear integer programming. Manag. Sci. 31, 1533–1546 (1985)CrossRefGoogle Scholar
  18. 18.
    Hadj-Alouane A.B., Bean J.C.: A genetic algorithm for the multiple choice integer program. Oper. Res. 45, 92–101 (1997)CrossRefGoogle Scholar
  19. 19.
    Homaifar A., Lai S.H.Y., Qi X.: Constrained optimization via genetic algorithms. Simulation 62, 242–254 (1994)CrossRefGoogle Scholar
  20. 20.
    Kaya C.Y., Noakes J.L.: A computational method for time-optimal control. J. Optim. Theor. Appl. 117, 69–92 (2003)CrossRefGoogle Scholar
  21. 21.
    Kennedy, J., Eberhart, R.: Particle swarm optimization. In: Proceedings of the IEEE international conference on neural networks, Piscataway, NJ, pp. 1942–1948 (1995)Google Scholar
  22. 22.
    Kirkpatrick S., Gelatt C.D., Vecchi M.P.: Optimization by simulated annealing. Science 220, 671–680 (1983)CrossRefGoogle Scholar
  23. 23.
    Manne, A.S.: GAMS/MINOS: Three examples. Technical report, Department of Operations Research, Stanford University, Stanford (1986)Google Scholar
  24. 24.
    Morales, A.K., Quezada, C.V.: A universal electic genetic algorithm for constraint optimization. In: Proceedings of the 6th european congress on intelligent techniques and soft computing, EUFIT’98, Aachen Germany, Verlag Mainz, pp. 518–522 (1998)Google Scholar
  25. 25.
    Michalewicz Z.: A survey of constraint handling techniques in evolutionary computation methods. In: McDonnell, J.R., Reynolds, R.G., Fogel, D.B. (eds) Proceedings of the 4th annual conference on evolutionary programming, pp. 135–155. MIT press, Cambridge (1995)Google Scholar
  26. 26.
    Sager, S.: mintOC, benchmark library of mixed-integer optimal control problems (http://mintoc.de) (2009)
  27. 27.
    Socha K., Dorigo M.: Ant colony optimization for continuous domains. Eur. J. Oper. Res. 185, 1155–1173 (2008)CrossRefGoogle Scholar
  28. 28.
    Schlüter M., Egea J.A., Banga J.R.: Extended antcolony optimization for non-convex mixed integer nonlinear programming. Comput. Oper. Res. 36(7), 2217–2229 (2009)CrossRefGoogle Scholar
  29. 29.
    Schlüter M., Egea J.A., Antelo L.T., Alonso A.A., Banga J.R.: An extended ant colony optimization algorithm for integrated process and control system design. Ind. Eng. Chem. 48(14), 6723–6738 (2009)CrossRefGoogle Scholar
  30. 30.
    Schlüter, M.: MIDACO—Global optimization software for mixed integer nonlinear programming (http://www.midaco-solver.com) (2009)
  31. 31.
    Smith A.E., Tate D.M.: Genetic optimization using a penalty function. In: Forrest, S. (eds) Proceedings of the 5th international conference on genetic algorithms, San Mateo, California, pp. 499–503. Morgan Kaufmann Publishers, Los Altos (1993)Google Scholar
  32. 32.
    Van de Braak, G.: Das Verfahren MISQP zur gemischt ganzzahligen nichtlinearen Programmierung fuer den Entwurf elektronischer Bauteile. Diploma Thesis, Department of Numerical and Instrumental Mathematics, University of Muenster, Germany (2001)Google Scholar
  33. 33.
    Yeniay O.: Penalty function methods for constrained optimization with genetic algorithms. Math. Comput. Appl. 10, 45–56 (2005)Google Scholar

Copyright information

© Springer Science+Business Media, LLC. 2009

Authors and Affiliations

  1. 1.Theoretical and Computational Optimization Group, School of MathematicsUniversity of BirminghamEdgbaston, BirminghamUK
  2. 2.Department of MathematicsUniversity of WürzburgWürzburgGermany

Personalised recommendations