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A filling function method for unconstrained global optimization

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Abstract

We consider the problem of finding a global minimum point of a given continuously differentiable function. The strategy is adopted of a sequential nonmonotone improvement of local optima. In particular, to escape the basin of attraction of a local minimum, a suitable Gaussian-based filling function is constructed using the quadratic model (possibly approximated) of the objective function, and added to the objective to fill the basin. Then, a procedure is defined where some new minima are determined, and that of them with the lowest function value is selected as the subsequent restarting point, even if its basin is higher than the starting one. Moreover, a suitable device employing repeatedly the centroid of all the minima determined, is introduced in order to improve the efficiency of the method in the solution of difficult problems where the number of local minima is very high. The algorithm is applied to a set of test functions from the literature and the numerical results are reported along with those obtained by applying a standard Monotonic Basin Hopping method for comparison.

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Notes

  1. Obviously, it is possible to define an algorithm where a further point is chosen at random and the above procedure is repeated. In this case, to avoid repetition of the same path, this further point is discarded if the new local minimum, determined by the local search routine, is near any one of the minima from which the procedure was already restarted.

  2. The final reached point \(\check{x}\) is considered the global solution if \(({f(\check{x})-f^*})/{\max \{1,|f^*|\}}\le 10^{-4}\).

References

  1. Addis, B., Cassioli, A., Locatelli, M., Schoen, F.: A global optimization method for the design of space trajectories. Comput. Optim. Appl. 48(3), 635–652 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  2. Bertsekas, D.P.: Constrained Optimization and Lagrange Multipliers Methods. Academic Press, NewYork (1982)

    Google Scholar 

  3. Byrd, R.H., Lu, P., Nocedal, J.: A limited memory algorithm for bound constrained optimization. SIAM J. Sci. Stat. Comput. 16(5), 1190–1208 (1995)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  Google Scholar 

  5. Dzemyda, G., S̆altenis, V., Z̆ilinskas, A. (eds.): Stochastic and Global Optimization. Nonconvex Optimization and its Applications. Kluwer, Dordrecht (2002)

    MATH  Google Scholar 

  6. Fang, A., O’Leary, D.: Modified Cholesky algorithms: a catalog with new approaches. Math. Program. 115, 319–349 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  7. Floudas, C.A., Pardalos, P.M., Adjiman, C.S., Esposito, W.R., Gumus, Z., Harding, S.T., Klepeis, J.L., Meyer, C.A., Schweiger, C.A.: Handbook of Test Problems for Local and Global Optimization. Kluwer, Dordrecht (1999)

    Book  Google Scholar 

  8. Gaviano, M., Kvasov, D.E., Lera, D., Sergeyev, YaD: Sergeyev. Algorithm 829: software for generation of classes of test functions with known local and global minima for global optimization. ACM Trans. Math. Softw. 29(4), 469–480 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  9. Ge, R.: A filled function method for finding a global minimizer of a function of several variables. Math. Program. 46, 191–204 (1990)

    Article  MATH  Google Scholar 

  10. Gribbin, J.: Almost Everyone’s Guide to Science: The Universe, Life and Everything. Orion Publishing Group Ltd., Phoenix (1999)

    Google Scholar 

  11. Hansen, P., Mladenović, N.: Variable neighborhood search: principles and applications. Eur. J. Oper. Res. 130(3), 449–467 (2001)

    Article  MATH  Google Scholar 

  12. Hedar, A.: http://www-optima.amp.i.kyoto-u.ac.jp/member/student/hedar/Hedar_files/TestGO.htm

  13. Horst, R., Pardalos, P.M., Thoai, V.N.: Introduction to Global Optimization. Nonconvex Optimization and Its Applications, 2nd edn. Kluwer, Dordrecht (2000)

    Book  MATH  Google Scholar 

  14. 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  MATH  MathSciNet  Google Scholar 

  15. Leary, R.H.: Global optimization on funneling landscapes. J. Glob. Optim. 18(4), 367–383 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  16. Levy, A.V., Montalvo, A.: The tunneling algorithm for the global minimization of functions. SIAM J. Sci. Stat. Comput. 6, 15–29 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  17. Liang, J.J., Qu, B-Y., Suganthan, P.N., Hernández-Díaz, A.G.: Problem definitions and evaluation criteria for the cec 2013 special session and competition on real-parameter optimization. Technical Report 201212, Computational Intelligence Laboratory, Zhengzhou University (2013)

  18. 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  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  20. 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  MATH  MathSciNet  Google Scholar 

  21. Locatelli, M.: On the multilevel structure of global optimization problems. Comput. Optim. Appl. 30(1), 5–22 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  22. Locatelli, M., Schoen, F.: Efficient algorithms for large scale global optimization: Lennard-Jones clusters. Comput. Optim. Appl. 26(2), 173–190 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  23. Locatelli, M., Schoen, F.: Global Optimization: Theory, Algorithms, and Applications. MOS-SIAM Series on Optimization. SIAM-MOS, Philadelphia (2013)

    Book  Google Scholar 

  24. Lucidi, S., Piccialli, V.: New classes of globally convexized filled functions for global optimization. J. Glob. Optim. 24(2), 219–236 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  25. Metropolis, N.: The beginning of the Monte Carlo method. Los Alamos Sci. 1987, 125–130 (1987)

    MathSciNet  Google Scholar 

  26. Mladenović, N., Hansen, P.: Variable neighborhood search. Comput. Oper. Res. 24(11), 1097–1100 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  27. Montaz Ali, M., Khompatraporn, C., Zabinsky, Z.B.: A numerical evaluation of several stochastic algorithms on selected continuous global optimization test problems. J. Glob. Optim. 31, 635–672 (2005)

    Article  MATH  Google Scholar 

  28. Pardalos, P.M., Romeijn, E. (eds.): Handbook of Global Optimization Volume 2. Series on Nonconvex Optimization and its Applications. Kluwer, Dordrecht (2002)

    Google Scholar 

  29. Pintér, J.D.: Global Optimization in Action. Continuous and Lipschitz Optimization: Algorithms, Implementations and Applications. Series on Nonconvex Optimization and Its Applications. Kluwer, Dordrecht (1996)

    Book  MATH  Google Scholar 

  30. Törn, A., Z̆ilinskas, A.: Global Optimization. Springer, Berlin (1989)

    Book  MATH  Google Scholar 

  31. Zhu, C., Byrd, R.H., Nocedal, J.: L-BFGS-B: Algorithm 778: L-BFGS-B, fortran routines for large scale bound constrained optimization. ACM Trans. Math. Softw. 23(4), 550–560 (1997)

    Article  MATH  MathSciNet  Google Scholar 

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Acknowledgments

This work has been partially supported by the FLAGSHIP “InterOmics” project (PB.P05) funded by the Italian MIUR and CNR organizations. We are indebted to the reviewers whose comments, criticisms and suggestions allowed us to significantly improve our work. Moreover, we thank Prof. Fabio Schoen, University of Florence, for providing a version of the MBH method.

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  1. Istituto di Analisi dei Sistemi ed Informatica “A. Ruberti”, Consiglio nazionale delle Ricerche, Viale Manzoni 30, 00185, Rome, Italy

    F. Lampariello & G. Liuzzi

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  1. F. Lampariello
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  2. G. Liuzzi
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Correspondence to G. Liuzzi.

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Lampariello, F., Liuzzi, G. A filling function method for unconstrained global optimization. Comput Optim Appl 61, 713–729 (2015). https://doi.org/10.1007/s10589-015-9728-6

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