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Journal of Heuristics

, Volume 22, Issue 4, pp 649–664 | Cite as

Flying elephants: a general method for solving non-differentiable problems

  • Adilson Elias Xavier
  • Vinicius Layter Xavier
Article

Abstract

Flying Elephants (FE) is a generalization and a new interpretation of the Hyperbolic Smoothing approach. The article introduces the fundamental smoothing procedures. It contains a general overview of successful applications of the approach for solving a select set of five important problems, namely: distance geometry, covering, clustering, Fermat–Weber and hub location. For each problem the original non-smooth formulation and the succedaneous completely differentiable one are presented. Computational experiments for all related problems obtained results that exhibited a high level of performance according to all criteria: consistency, robustness and efficiency. For each problem some results to illustrate the performance of FE are also presented.

Keywords

Non-differentiable optimization Smoothing Distance geometry Covering Clustering Fermat–Weber problem Hub location problem 

Notes

Acknowledgments

The authors would like to thank the two referees for their comments.

References

  1. Bagirov, A.M., Al-Nuaimat, A., Sultanova, N.: Hyperbolic smoothing function method for minimax problems. Optimization 62(6), 759–782 (2013)Google Scholar
  2. Bagirov, A.M., Ordin, B., Ozturk, G., Xavier, A.E.: An incremental clustering algorithm based on smoothing techniques. Submitted to Computational Optimization and Applications (2012)Google Scholar
  3. Brimberg, J., Hansen, P., Mladenovic, N., Taillard, E.D.: Improvements and comparison of heuristics for solving the multisource weber problem. Oper. Res. 48, 129–135 (2000)MathSciNetCrossRefGoogle Scholar
  4. Chaves, A.M.V.: Resolução do problema minimax via suavização. M.Sc. Thesis—COPPE-UFRJ, Rio de Janeiro (1997)Google Scholar
  5. Contreras, I., Cordeau, J.-F., Laporte, G.: Benders decomposition for large-scale uncapacitated hub location. Oper. Res. 59, 1477–1490 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  6. Demyanov, V.F., Malozemov, V.N.: Introduction to Minimax. Dover Publications Inc., New York (1974)Google Scholar
  7. Demyanov, V.F., Pardalos, P.M., Batsyn, M.: Constructive Nonsmooth Analysis and Related Topics. Springer, New York (2014)CrossRefzbMATHGoogle Scholar
  8. Du, D.Z., Pardalos, P.M.: Minimax and Applications. Kluwer Academic Press, Dordrecht (1995)CrossRefzbMATHGoogle Scholar
  9. Gesteira, C.: Resolvendo Problemas de Localização de Hubs com Alocação Múltipla numa Modelagem Contínua Tipo p-Medianas Usando a Abordagem de Suavização Hiperbólica. M.Sc. Thesis—COPPE-UFRJ, Rio de Janeiro (2012)Google Scholar
  10. Hoai An, L.T., Tao, P.D.: Large-Scale Molecular Conformation Via the Exact Distance Geometry Problem. Lecture Notes in Economics and Mathematical Systems, vol. 481, pp. 260–277 (2000)Google Scholar
  11. Macambira, A.F.U.S.: Determinação de estruturas de proteínas via suavização e penalização hiperbólica. M.Sc. Thesis—COPPE—UFRJ, Rio de Janeiro (2003)Google Scholar
  12. Moré, J.J., Wu, Z.: \(\epsilon -\)optimal solutions to distance geometry problems via global continuation. Mathematics and Computer Science Division, Argonne Lab., Preprint MCS-P520-0595 (1995)Google Scholar
  13. Pardalos, P.M., Shalloway, D., Xue, G.: Global Minimization of Nonconvex Energy Functions: Molecular Conformation and Protein Folding DIMACS Series, vol. 23. American Mathematical Society, Providence (1996)zbMATHGoogle Scholar
  14. Plastino, A., Fuchshuber, R., Martins, S.L., Freitas, A.A., Salhi, S.: A hybrid data mining metaheuristic for the \(p-\)median problem. Statist. Anal. Data Mining 4, 313–335 (2011)MathSciNetCrossRefGoogle Scholar
  15. Rubinov, A.M.: Methods for global optimization of nonsmooth functions with applications. Appl. Comput. Math. 5, 3–15 (2006)MathSciNetzbMATHGoogle Scholar
  16. Santos, A.B.A.: Problemas de programação não-diferenciável: Uma metodologia de suavização. M.Sc. Thesis—COPPE—UFRJ, Rio de Janeiro (2003)Google Scholar
  17. Souza, M.F.: Suavização hiperbólica aplicada à otimização de geometria molecular. D.Sc. Thesis—COPPE—UFRJ, Rio de Janeiro (2010)Google Scholar
  18. Souza, M.F., Xavier, A.E., Lavor, C., Maculan, N.: Hyperbolic smoothing and penalty techniques applied to molecular structure determination. Oper. Res. Lett. 39, 461–465 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  19. Wei, H.: Solving continuous space location problems. Ph.D. Thesis—The Ohio State University (2008)Google Scholar
  20. Xavier, A.E.: Penalização hiperbólica: Um novo método para resolução de problemas de otimização. M.Sc. Thesis—COPPE—UFRJ, Rio de Janeiro (1982)Google Scholar
  21. Xavier, A.E.: Convexificação do problema de distância geométrica através da técnica de suavização hiperbólica. Workshop em Biociências COPPE UFRJ (2003)Google Scholar
  22. Xavier, A.E.: The hyperbolic smoothing clustering method. Pattern Recogn. 43, 731–737 (2010)CrossRefzbMATHGoogle Scholar
  23. Xavier, V.L.: Resolução do Problema de Agrupamento segundo o Critério de Minimização da Soma das Distâncias. M.Sc. Thesis—COPPE—UFRJ, Rio de Janeiro (2012)Google Scholar
  24. Xavier, A.E., Fernandes Oliveira, A.A.: Optimal covering of plane domains by circles via hyperbolic smoothing. J. Global Optim. 31, 493–504 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  25. Xavier, V.L., França, F.M.G., Xavier, A.E., Lima, P.M.V.: A hyperbolic smoothing approach to the Multisource Weber problem. J. Global Optim. 60(1), 49–58 (2014a)Google Scholar
  26. Xavier, A.E., Gesteira, C., Xavier, V.L.: The continuous multiple allocation p-hub median problem solving by the hyperbolic smoothing approach: computational performance. Optimization (2014b). doi: 10.1080/02331934.2014.929677
  27. Xavier, A.E., Xavier, V.L.: Solving the minimum sum-of-squares clustering problem by hyperbolic smoothing and partition into boundary and gravitational regions. Pattern Recogn. 44, 70–77 (2011)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Adilson Elias Xavier
    • 1
  • Vinicius Layter Xavier
    • 1
  1. 1.Federal University of Rio de JaneiroRio de JaneiroBrazil

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