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Eurofuse 2011 pp 293-301 | Cite as

Penalty Fuzzy Function for Derivative-Free Optimization

  • J. Matias
  • P. Mestre
  • A. Correia
  • P. Couto
  • C. Serodio
  • P. Melo-Pinto
Part of the Advances in Intelligent and Soft Computing book series (AINSC, volume 107)

Abstract

Penalty and Barrier methods are normally used to solve Nonlinear Optimization Constrained Problems. The problems appear in areas such as engineering and are often characterized by the fact that involved functions (objective and constraints) are non-smooth and/or their derivatives are not know. This means that optimization methods based on derivatives cannot be used. A Java based API was implemented, including only derivative-free optimization methods, to solve both constrained and unconstrained problems, which includes Penalty and Barriers methods. In this work a new penalty function, based on Fuzzy Logic, is presented. This function imposes a progressive penalization to solutions that violate the constraints. This means that the function imposes a low penalization when the violation of the constraints is low and a heavy penalization when the violation is high. The value of the penalization is not known in beforehand, it is the outcome of a fuzzy inference engine. Numerical results comparing the proposed function with two of the classic penalty/barrier functions are presented. Regarding the presented results one can conclude that the proposed penalty function besides being very robust also exhibits a very good performance.

Keywords

Penalty Function Unconstrained Optimization Problem Barrier Method Direct Search Method Mesh Adaptive Direct Search 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Audet, C.: Convergence results for pattern search algorithms are tight. Optimization and Engineering 2(5), 101–122 (2004)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Audet, C., Bchard, V., Digabel, S.L.: Nonsmooth optimization through mesh adaptive direct search and variable neighborhood search. J. Global Opt. (41), 299–318 (2008)zbMATHCrossRefGoogle Scholar
  3. 3.
    Audet, C., Dennis, J.: A pattern search filter method for nonlinear programming without derivatives. SIAM Journal on Optimization 5(14), 980–1010 (2004)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Audet, C., Dennis Jr., J.E.: Analysis of generalized pattern searches. SIAM Journal on Optimization 13(3), 889–903 (2002)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Audet, C., Dennis Jr., J.E.: Mesh adaptive direct search algorithms for constrained optimization. SIAM Journal on Optimization (17), 188–217 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Audet, C., Dennis, Jr., J.E.: A mads algorithm with a progressive barrier for derivative-free nonlinear programming. Tech. Rep. G-2007-37, Les Cahiers du GERAD, cole Polytechnique de Montral (2007)Google Scholar
  7. 7.
    Audet, C., Dennis Jr, J.E., Digabel, S.L.: Globalization strategies for mesh adaptative direct search.Tech. Rep. G-2008-74, Les Cahiers du GERAD, cole Polytechnique de Montral (2008)Google Scholar
  8. 8.
    Bongartz, I., Conn, A., Gould, N., Toint, P.: Cute: Constrained and unconstrained testing environment. ACM Transactions and Mathematical Software (21), 123–160 (1995)zbMATHCrossRefGoogle Scholar
  9. 9.
    Byrd, R.H., Nocedal, J., Waltz, R.A.: Steering exact penalty methods for nonlinear programming. Optimization Methods & Software 23(2), 197–213 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Conn, A.R., Scheinberg, K., Vicente, L.N.: Introduction to Derivative-Free Optimization. MPS-SIAM Series on Optimization. SIAM, Philadelphia (2009)zbMATHCrossRefGoogle Scholar
  11. 11.
    Correia, A., Matias, J., Mestre, P., Serdio, C.: Direct-search penalty/barrier methods. In: World Congress on Engineering 2010. Lecture Notes in Engineering and Computer Science, vol. 3, pp. 1729–1734. IAENG, London (2010)Google Scholar
  12. 12.
    Gould, N.I.M., Orban, D., Toint, P.L.: An interior-point l 1-penalty method for nonlinear optimization. Tech. rep., Rutherford Appleton Laboratory Chilton (2003)Google Scholar
  13. 13.
    Hooke, R., Jeeves, T.: Direct search solution of numerical and statistical problems. Journal of the Association for Computing Machinery 8(2), 212–229 (1961)zbMATHGoogle Scholar
  14. 14.
    Kolda, T., Lewis, R., Torczon, V.: Optimization by direct search: New perspectives on some classical and modern methods. SIAM Review 45, 385–482 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Lewis, R., Torczon, V., Trosset, M.: Direct search methods: Then and now. J. Comput. Appl. Math. (124), 191–207 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Pietrzykowski, T.: An exact potential method for constrained maxima. SIAM Journal on Numerical Analysis 6(2), 299–304 (1969)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Schittkowski, K.: More Test Examples for Nonlinear Programming Codes, Economics and Mathematical Systems. Springer, Heidelberg (1987)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • J. Matias
    • 1
  • P. Mestre
    • 1
  • A. Correia
    • 2
  • P. Couto
    • 1
  • C. Serodio
    • 1
  • P. Melo-Pinto
    • 1
  1. 1.CM-UTADVila RealPortugal
  2. 2.CM-UTAD and CIICESI/ESTGF/IPPFelgueirasPortugal

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