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A DIRECT-type approach for derivative-free constrained global optimization

Computational Optimization and Applications volume 65, pages 361–397 (2016)Cite this article

Abstract

In the field of global optimization, many efforts have been devoted to globally solving bound constrained optimization problems without using derivatives. In this paper we consider global optimization problems where both bound and general nonlinear constraints are present. To solve this problem we propose the combined use of a DIRECT-type algorithm with a derivative-free local minimization of a nonsmooth exact penalty function. In particular, we define a new DIRECT-type strategy to explore the search space by explicitly taking into account the two-fold nature of the optimization problems, i.e. the global optimization of both the objective function and of a feasibility measure. We report an extensive experimentation on hard test problems to show viability of the approach.

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References

  1. Birgin, E.G., Floudas, C.A., Martinez, J.M.: Global minimization using an augmented lagrangian method with variable lower-level constraints. Math. Program. (Ser. A) 125(1), 139–162 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  2. Campana, E.F., Liuzzi, G., Lucidi, S., Peri, D., Pinto, A., Piccialli, V.: New global optimization methods for ship design problems. Optim. Eng. 10(4), 533–555 (2009)

    Article  MATH  Google Scholar 

  3. 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)

    Article  Google Scholar 

  4. Di Pillo, G., Lucidi, S., Rinaldi, F.: An approach to constrained global optimization based on exact penalty functions. J. Glob. Optim. 54(2), 251–260 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  5. Di Pillo, G., Lucidi, S., Rinaldi, F.: A derivative-free algorithm for constrained global optimization based on exact penalty functions. J. Optim. Theory Appl. (2013)

  6. Fasano, G., Liuzzi, G., Lucidi, S., Rinaldi, F.: A linesearch-based derivative-free approach for nonsmooth constrained optimization. SIAM J. Optim. (2014)

  7. Gablonsky, J.M.: DIRECT Version 2.0, User Guide

  8. Hansen, N., Müller, S.D., Koumoutsakos, P.: Reducing the time complexity of the derandomized evolution strategy with covariance matrix adaptation (CMA-ES). Evol. Comput. 11(1), 1–18 (2003)

    Article  Google Scholar 

  9. Hansen, N., Ostermeier, A.: Completely derandomized self-adaptation in evolution strategies. Evol. Comput. 9(2), 159–195 (2001)

    Article  Google Scholar 

  10. Jones, D.R.: The direct global optimization algorithm. In: Floudas, C., Pardalos, P. (eds.) Encyclopedia of Optimization, pp. 431–440. Kluwer Academic Publishers, Dordrecht (2001)

    Chapter  Google Scholar 

  11. Jones, D.R., Perttunen, C.D., Stuckman, B.E.: Lipschitzian optimization without the lipschitz constant. J. Optim. Theory Appl. 79(1), 157–181 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  12. 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(2), 353–375 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  13. Liuzzi, G., Lucidi, S., Piccialli, V.: A partition-based global optimization algorithm. J. Glob. Optim. 48, 113–128 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  14. Liuzzi, G., Lucidi, S., Piccialli, V.: Partitioning techniques for global optimization. In De Simone, V., Di Serafino, D., Toraldo, G. (eds.) Recent Advances in Nonlinear Optimization and Equilibrium Problems: A Tribute to Marco D’Apuzzo, volume 27 of Quaderni di Matematica, Dipartimento di Matematica, Seconda Università di Napoli, pp. 253–272. Aracne Editrice S.r.l. (2012)

  15. Liuzzi, G., Lucidi, S., Piccialli, V.: Exploiting derivative-free local searches in direct-type algorithms for global optimization. Comput. Optim. Appl., 1–27 (2015)

  16. Liuzzi, G., Lucidi, S., Piccialli, V., Sotgiu, A.: A magnetic resonance device designed via global optimization techniques. Math. Program. 101(2), 339–364 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  17. Sergeyev, Y.D.: On convergence of divide the best global optimization algorithms. Optimization 44, 303–325 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  18. Shcherbina, O., Neumaier, A., Sam-Haroud, D., Vu, X.H., Nguyen, T.V.: Benchmarking global optimization and constraint satisfaction codes. In: Bliek, Ch., Jermann, Ch., Neumaier, A. (eds.) Global Optimization and Constraint Satisfaction, pp. 211–222. Springer, Berlin (2003)

    Chapter  Google Scholar 

  19. Tawarmalani, M., Sahinidis, N.V.: Convexification and global optimization in continuous and mixed-integer nonlinear programming. In: Nonconvex Optimization and its Applications. Kluwer Academic Publishers, Dordrecht (2002)

  20. Wang, C.Y., Li, D.: Unified theory of augmented lagrangian methods for constrained global optimization. J. Glob. Optim. 44, 433–458 (2009)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgments

This work has been partially funded by the UE (ENIAC Joint Undertaking) in the MODERN project (ENIAC-120003).

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Authors and Affiliations

  1. Istituto di Analisi dei Sistemi ed Informatica “A. Ruberti”, Consiglio Nazionale delle Ricerche, Via dei Taurini 19, 00185, Rome, Italy

    G. Liuzzi

  2. Department of Computer, Control and Management Engineering, “Sapienza” University of Rome, via Ariosto 25, 00185, Rome, Italy

    G. Di Pillo & S. Lucidi

  3. Dipartimento di Ingegneria Civile e Ingegneria Informatica, Università degli Studi di Roma “Tor Vergata”, Viale del Politecnico 1, 00133, Rome, Italy

    V. Piccialli

  4. Dipartimento di Matematica, Università di Padova, Via Trieste 63, 35121, Padua, Italy

    F. Rinaldi

Authors
  1. G. Di Pillo
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  2. G. Liuzzi
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  3. S. Lucidi
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  4. V. Piccialli
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  5. F. Rinaldi
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Correspondence to F. Rinaldi.

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Di Pillo, G., Liuzzi, G., Lucidi, S. et al. A DIRECT-type approach for derivative-free constrained global optimization. Comput Optim Appl 65, 361–397 (2016). https://doi.org/10.1007/s10589-016-9876-3

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  • DOI: https://doi.org/10.1007/s10589-016-9876-3

Keywords

  • Global optimization
  • Derivative-free optimization
  • Nonlinear optimization
  • DIRECT-type algorithm