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


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.


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



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


  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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