Computational Optimization and Applications

, Volume 71, Issue 1, pp 53–72 | Cite as

A multi-objective DIRECT algorithm for ship hull optimization

  • E. F. Campana
  • M. Diez
  • G. LiuzziEmail author
  • S. Lucidi
  • R. Pellegrini
  • V. Piccialli
  • F. Rinaldi
  • A. Serani


The paper is concerned with black-box nonlinear constrained multi-objective optimization problems. Our interest is the definition of a multi-objective deterministic partition-based algorithm. The main target of the proposed algorithm is the solution of a real ship hull optimization problem. To this purpose and in pursuit of an efficient method, we develop an hybrid algorithm by coupling a multi-objective DIRECT-type algorithm with an efficient derivative-free local algorithm. The results obtained on a set of “hard” nonlinear constrained multi-objective test problems show viability of the proposed approach. Results on a hull-form optimization of a high-speed catamaran (sailing in head waves in the North Pacific Ocean) are also presented. In order to consider a real ocean environment, stochastic sea state and speed are taken into account. The problem is formulated as a multi-objective optimization aimed at (i) the reduction of the expected value of the mean total resistance in irregular head waves, at variable speed and (ii) the increase of the ship operability, with respect to a set of motion-related constraints. We show that the hybrid method performs well also on this industrial problem.


Multi-objective nonlinear programming Derivative-free optimization DIRECT-type algorithm 

Mathematics Subject Classification

90C30 90C56 65K05 



We thank two anonymous reviewers whose comments helped us improve the paper.


  1. 1.
    Bandyopadhyay, S., Pal, S.K., Aruna, B.: Multiobjective GAs, quantitative indices, and pattern classification. Syst. Man Cybern. B IEEE Trans. Cybern. 34(5), 2088–2099 (2004)CrossRefGoogle Scholar
  2. 2.
    Campana, E.F., Diez, M., Iemma, U., Liuzzi, G., Lucidi, S., Rinaldi, F., Serani, A.: Derivative-free global ship design optimization using global/local hybridization of the DIRECT algorithm. Optim. Eng. 17(1), 127–156 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Chen, X., Diez, M., Kandasamy, M., Campana, E.F., Stern, F.: Design optimization of the waterjet-propelled delft catamaran in calm water using urans, design of experiments, metamodels and swarm intelligence. In: Proceedings of the 12th International Conference on Fast Sea Transportation (FAST2013), Amsterdam, The Netherlands, pp. 1–12 (2013)Google Scholar
  4. 4.
    Chen, X., Diez, M., Kandasamy, M., Zhang, Z., Campana, E.F., Stern, F.: High-fidelity global optimization of shape design by dimensionality reduction, metamodels and deterministic particle swarm. Eng. Optim. 47(4), 473–494 (2015)CrossRefGoogle Scholar
  5. 5.
    Conn, A., Scheinberg, K., Vicente, L.N.: Introduction to Derivative-Free Optimization, vol. 8. SIAM, New Delhi (2009)CrossRefzbMATHGoogle Scholar
  6. 6.
    Custódio, A.L., Madeira, J.F.A., Vaz, A.I.F., Vicente, L.N.: Direct multisearch for multiobjective optimization. SIAM J. Optim. 21(3), 1109–1140 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Deb, K., Pratap, A., Agarwal, S., Meyarivan, T.: A fast and elitist multiobjective genetic algorithm: NSGA-II. IEEE Trans. Evol. Comput. 6(2), 182–197 (2002)CrossRefGoogle Scholar
  8. 8.
    Di Pillo, G., Liuzzi, G., Lucidi, S., Piccialli, V., Rinaldi, F.: A direct-type approach for derivative-free constrained global optimization. Comput. Optim. Appl. 65(2), 361–397 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Diez, M., Campana, E.F., Stern, F.: Design-space dimensionality reduction in shape optimization by Karhunen–Loève expansion. Comput. Methods Appl. Mech. Eng. 283, 1525–1544 (2015)CrossRefGoogle Scholar
  10. 10.
    Diez, M., Chen, X., Campana, E.F., Stern, F.: Reliability-based robust design optimization for ships in real ocean environment. In: Proceedings of the 12th International Conference on Fast Sea Transportation (FAST2013), Amsterdam, The Netherlands, pp. 1–17 (2013)Google Scholar
  11. 11.
    Dolan, E.D., Moré, J.J.: Benchmarking optimization software with performance profiles. Math. Program. 91(2), 201–213 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Ehrgott, M.: Multicriteria Optimization, 2nd edn. Springer, Berlin (2005)zbMATHGoogle Scholar
  13. 13.
    Evtushenko, Y.G., Posypkin, M.A.: Nonuniform covering method as applied to multicriteria optimization problems with guaranteed accuracy. Comput. Math. Math. Phys. 53(2), 144–157 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Evtushenko, Y.G., Posypkin, M.A.: A deterministic algorithm for global multi-objective optimization. Optim. Methods Softw. 29(5), 1005–1019 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Gablonsky, J.M., Kelley, C.T.: A locally-biased form of the DIRECT algorithm. J. Global Optim. 21(1), 27–37 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Gen, M., Cheng, R., Lin, L.: Multiobjective Genetic Algorithms, pp. 1–47. Springer, Berlin (2008)Google Scholar
  17. 17.
    He, J., Verstak, A., Watson, L.T., Sosonkina, M.: Design and implementation of a massively parallel version of DIRECT. Comput. Optim. Appl. 40, 217–245 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    He, W., Diez, M., Zou, Z., Campana, E.F., Stern, F.: URANS study of Delft catamaran total/added resistance, motions and slamming loads in head sea including irregular wave and uncertainty quantification for variable regular wave and geometry. Ocean Eng. 74, 189–217 (2013)CrossRefGoogle Scholar
  19. 19.
    Jones, D.R., Perttunen, C.D., Stuckman, B.E.: Lipschitzian optimization without the Lipschitz constant. J. Optim. Theory Appl. 79(1), 157–181 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Kvasov, D.E., Sergeyev, Y.D.: Deterministic approaches for solving practical black-box global optimization problems. Adv. Eng. Softw. 80, 58–66 (2015)CrossRefGoogle Scholar
  21. 21.
    Lera, D., Sergeyev, Y.D.: Deterministic global optimization using space-filling curves and multiple estimates of Lipschitz and Hölder constants. Commun. Nonlinear Sci. Numer. Simul. 23(1–3), 328–342 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Liu, Q., Zeng, J.: Global optimization by multilevel partition. J. Global Optim. 61(1), 47–69 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Liuzzi, G., Lucidi, S., Piccialli, V.: A DIRECT-based approach exploiting local minimizations for the solution of large-scale global optimization problems. Comput. Optim. Appl. 45, 353–375 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Liuzzi, G., Lucidi, S., Piccialli, V.: A partition-based global optimization algorithm. J. Global Optim. 48, 113–128 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Liuzzi, G., Lucidi, S., Piccialli, V.: Exploiting derivative-free local searches in direct-type algorithms for global optimization. Comput. Optim. Appl. 65(2), 449–475 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Liuzzi, G., Lucidi, S., Rinaldi, F.: A derivative-free approach to constrained multiobjective nonsmooth optimization. SIAM J. Optim. 26(4), 2744–2774 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Miettinen, K.: Nonlinear Multiobjective Optimization. International Series in Operations Research and Management Science. Springer, Berlin (1998)CrossRefGoogle Scholar
  28. 28.
    Paulavičius, R., Sergeyev, Y.D., Kvasov, D.E., Žilinskas, J.: Globally-biased disimpl algorithm for expensive global optimization. J. Global Optim. 59(2–3), 545–567 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Sergeyev, Y.D.: On convergence of “Divide the Best” global optimization algorithms. Optimization 44(3), 303–325 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Sergeyev, Y.D., Kvasov, D.E.: Global search based on efficient diagonal partitions and a set of Lipschitz constants. SIAM J. Optim. 16(3), 910–937 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Strongin, R.G., Sergeyev, Y.D.: Global Optimization with Non-Convex Constraints: Sequential and Parallel Algorithms. Kluwer, Dordrecht (2000)CrossRefzbMATHGoogle Scholar
  32. 32.
    Volpi, S., Diez, M., Gaul, N.J., Song, H., Iemma, U., Choi, K.K., Campana, E.F., Stern, F.: Development and validation of a dynamic metamodel based on stochastic radial basis functions and uncertainty quantification. Struct. Multidiscip. Optim. 51(2), 347–368 (2015)CrossRefGoogle Scholar
  33. 33.
    Žilinskas, A., Gimbutienė, G.: On one-step worst-case optimal trisection in univariate bi-objective lipschitz optimization. Commun. Nonlinear Sci. Numer. Simul. 35, 123–136 (2016)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Žilinskas, A., Žilinskas, J.: Adaptation of a one-step worst-case optimal univariate algorithm of bi-objective lipschitz optimization to multidimensional problems. Commun. Nonlinear Sci. Numer. Simul. 21(1), 89–98 (2015). Numerical Computations: Theory and Algorithms (NUMTA 2013), International Conference and Summer SchoolGoogle Scholar
  35. 35.
    Zhou, A., Qu, B.-Y., Li, H., Zhao, S.-Z., Suganthan, P.N., Zhang, Q.: Multiobjective evolutionary algorithms: a survey of the state of the art. Swarm Evolut. Comput. 1(1), 32–49 (2011)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Istituto Nazionale per Studi ed Esperienze di Architettura NavaleConsiglio Nazionale delle RicercheRomeItaly
  2. 2.Istituto di Analisi dei Sistemi ed InformaticaConsiglio Nazionale delle RicercheRomeItaly
  3. 3.Dipartimento di Ingegneria Informatica, Automatica e GestionaleSapienza Università di RomaRomeItaly
  4. 4.Dipartimento di Ingegneria Civile e Ingegneria InformaticaUniversità degli Studi di Roma “Tor Vergata”RomeItaly
  5. 5.Dipartimento di MatematicaUniversità di PadovaPaduaItaly

Personalised recommendations