Advertisement

Journal of Global Optimization

, Volume 59, Issue 2–3, pp 545–567 | Cite as

Globally-biased Disimpl algorithm for expensive global optimization

  • Remigijus Paulavičius
  • Yaroslav D. SergeyevEmail author
  • Dmitri E. Kvasov
  • Julius Žilinskas
Article

Abstract

Direct-type global optimization algorithms often spend an excessive number of function evaluations on problems with many local optima exploring suboptimal local minima, thereby delaying discovery of the global minimum. In this paper, a globally-biased simplicial partition Disimpl algorithm for global optimization of expensive Lipschitz continuous functions with an unknown Lipschitz constant is proposed. A scheme for an adaptive balancing of local and global information during the search is introduced, implemented, experimentally investigated, and compared with the well-known Direct and Direct l methods. Extensive numerical experiments executed on 800 multidimensional multiextremal test functions show a promising performance of the new acceleration technique with respect to competitors.

Keywords

Global optimization Lipschitz condition Direct algorithm  Two-phase approach Globally-biased Disimpl algorithm 

Notes

Acknowledgments

The authors would like to thank anonymous referees for their careful reading of the paper and insightful comments that helped us to improve the paper. Postdoctoral fellowship of R. Paulavičius is being funded by European Union Structural Funds project “Postdoctoral Fellowship Implementation in Lithuania” within the framework of the Measure for Enhancing Mobility of Scholars and Other Researchers and the Promotion of Student Research (VP1-3.1-ŠMM-01) of the Program of Human Resources Development Action Plan. The research work of Ya. D. Sergeyev and D. E. Kvasov was partially supported by the INdAM–GNCS 2014 Research Project of the Italian National Group for Scientific Computation of the National Institute for Advanced Mathematics “F. Severi”. The closing part of this research has been done in the framework of the project “Multiextremal optimization: Efficient global search algorithms and supercomputing” submitted to the Russian Scientific Fund.

References

  1. 1.
    Baker, C.A., Watson, L.T., Grossman, B., Mason, W.H., Haftka, R.T.: Parallel global aircraft configuration design space exploration. In: A. Tentner (ed.) High Performance Computing Symposium 2000, pp. 54–66. Soc. for Computer Simulation Internat (2000)Google Scholar
  2. 2.
    Bartholomew-Biggs, M.C., Parkhurst, S.C., Wilson, S.P.: Using DIRECT to solve an aircraft routing problem. Comput. Optim. Appl. 21(3), 311–323 (2002). doi: 10.1023/A:1013729320435 CrossRefGoogle Scholar
  3. 3.
    Carter, R.G., Gablonsky, J.M., Patrick, A., Kelley, C.T., Eslinger, O.J.: Algorithms for noisy problems in gas transmission pipeline optimization. Optim. Eng. 2(2), 139–157 (2001). doi: 10.1023/A:1013123110266 CrossRefGoogle Scholar
  4. 4.
    Casado, L.G., García, I., Tóth-G, B., Hendrix, E.M.T.: On determining the cover of a simplex by spheres centered at its vertices. J. Global Optim. 50(4), 645–655 (2011). doi: 10.1007/s10898-010-9524-x CrossRefGoogle Scholar
  5. 5.
    Cox, S.E., Haftka, R.T., Baker, C.A., Grossman, B., Mason, W.H., Watson, L.T.: A comparison of global optimization methods for the design of a high-speed civil transport. J. Global Optim. 21(4), 415–432 (2001). doi: 10.1023/A:1012782825166 CrossRefGoogle Scholar
  6. 6.
    Custódio, A.L., Rocha, H., Vicente, L.N.: Incorporating minimum Frobenius norm models in direct search. Comput. Optim. Appl. 46(2), 265–278 (2010). doi: 10.1007/s10589-009-9283-0 CrossRefGoogle Scholar
  7. 7.
    Di Serafino, D., Liuzzi, G., Piccialli, V., Riccio, F., Toraldo, G.: A modified DIviding RECTangles algorithm for a problem in astrophysics. J. Optim. Theory Appl. 151(1), 175–190 (2011). doi: 10.1007/s10957-011-9856-9 CrossRefGoogle Scholar
  8. 8.
    Dixon, L.C.W., Szegö, G.P. (eds.): Towards Global Optimisation, vol. 2. North-Holland Publishing Company, Amsterdam (1978)Google Scholar
  9. 9.
    Dolan, E.D., Moré, J.J.: Benchmarking optimization software with performance profiles. Math. Program. 91(2), 201–213 (2002). doi: 10.1007/s101070100263 CrossRefGoogle Scholar
  10. 10.
    Elsakov, S.M., Shiryaev, V.I.: Homogeneous algorithms for multiextremal optimization. Comput. Math. Math. Phys. 50(10), 1642–1654 (2010). doi: 10.1134/S0965542510100027 CrossRefGoogle Scholar
  11. 11.
    Evtushenko, Y.G., Posypkin, M.A.: A deterministic approach to global box-constrained optimization. Optim. Lett. 7(4), 819–829 (2013). doi: 10.1007/s11590-012-0452-1 CrossRefGoogle Scholar
  12. 12.
    Finkel, D.E.: Global Optimization with the Direct Algorithm. Ph.D. thesis, North Carolina State University (2005)Google Scholar
  13. 13.
    Finkel, D.E., Kelley, C.T.: Additive scaling and the DIRECT algorithm. J. Global Optim. 36(4), 597–608 (2006). doi: 10.1007/s10898-006-9029-9 CrossRefGoogle Scholar
  14. 14.
    Floudas, C.A., Pardalos, P.M. (eds.): Encyclopedia of Optimization, vol. 6, 2nd edn. Springer, Berlin (2009)Google Scholar
  15. 15.
    Gablonsky, J.M.: Modifications of the Direct algorithm. Ph.D. thesis, North Carolina State University (2001)Google Scholar
  16. 16.
    Gablonsky, J.M., Kelley, C.T.: A locally-biased form of the DIRECT algorithm. J. Global Optim. 21(1), 27–37 (2001). doi: 10.1023/A:1017930332101 CrossRefGoogle Scholar
  17. 17.
    Gaviano, M., Kvasov, D.E., Lera, D., Sergeyev, Y.D.: 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). doi: 10.1145/962437.962444 CrossRefGoogle Scholar
  18. 18.
    Gorodetsky, S.Y.: Paraboloid triangulation methods in solving multiextremal optimization problems with constraints for a class of functions with Lipschitz directional derivatives. Vestnik Lobachevsky State Univ. Nizhni Novgorod 1(1), 144–155 (2012). In RussianGoogle Scholar
  19. 19.
    Gorodetsky, S.Y.: Several approaches to generalization of the DIRECT method to problems with functional constraints. Vestnik of Lobachevsky State Univ. Nizhni Novgorod 6(1), 189–215 (2013). In RussianGoogle Scholar
  20. 20.
    Grbić, R., Nyarko, E.K., Scitovski, R.: A modification of the direct method for Lipschitz global optimization for a symmetric function. J. Global Optim. 57(4), 1193–1212 (2013). doi: 10.1007/s10898-012-0020-3 CrossRefGoogle Scholar
  21. 21.
    Grishagin, V.A.: Operating characteristics of some global search algorithms. In: Problems of Stochastic Search, vol. 7, pp. 198–206. Zinatne, Riga (1978). In RussianGoogle Scholar
  22. 22.
    He, J., Watson, L.T., Ramakrishnan, N., Shaffer, C.A., Verstak, A., Jiang, J., Bae, K., Tranter, W.H.: Dynamic data structures for a DIRECT search algorithm. Comput. Optim. Appl. 23(1), 5–25 (2002). doi: 10.1023/A:1019992822938 CrossRefGoogle Scholar
  23. 23.
    Horst, R., Pardalos, P.M. (eds.): Handbook of Global Optimization, vol. 1. Kluwer Academic Publishers, Dordrech (1995)Google Scholar
  24. 24.
    Horst, R., Tuy, H.: Global Optimization: Deterministic Approaches. Springer, Berlin (1996)CrossRefGoogle Scholar
  25. 25.
    Jones, D.R.: The direct global optimization algorithm. In: Floudas, C.A., Pardalos, P.M. (eds.) The Encyclopedia of Optimization, pp. 431–440. Kluwer Academic Publishers, Dordrect (2001)CrossRefGoogle Scholar
  26. 26.
    Jones, D.R., Perttunen, C.D., Stuckman, B.E.: Lipschitzian optimization without the Lipschitz constant. J. Optim. Theory Appl. 79(1), 157–181 (1993). doi: 10.1007/BF00941892 CrossRefGoogle Scholar
  27. 27.
    Kvasov, D.E., Pizzuti, C., Sergeyev, Y.D.: Local tuning and partition strategies for diagonal GO methods. Numer. Math. 94(1), 93–106 (2003). doi: 10.1007/s00211-002-0419-8 CrossRefGoogle Scholar
  28. 28.
    Kvasov, D.E., Sergeyev, Y.D.: A univariate global search working with a set of Lipschitz constants for the first derivative. Optim. Lett. 3(2), 303–318 (2009). doi: 10.1007/s11590-008-0110-9 CrossRefGoogle Scholar
  29. 29.
    Kvasov, D.E., Sergeyev, Y.D.: Lipschitz gradients for global optimization in a one-point-based partitioning scheme. J. Comput. Appl. Math. 236(16), 4042–4054 (2012). doi: 10.1016/j.cam.2012.02.020 CrossRefGoogle Scholar
  30. 30.
    Kvasov, D.E., Sergeyev, Y.D.: Univariate geometric Lipschitz global optimization algorithms. Numer. Algebr. Control Optim. 2(1), 69–90 (2012). doi: 10.3934/naco.2012.2.69 CrossRefGoogle Scholar
  31. 31.
    Kvasov, D.E., Sergeyev, Y.D.: Lipschitz global optimization methods in control problems. Autom. Remote Control 74(9), 1435–1448 (2013). doi: 10.1134/S0005117913090014 CrossRefGoogle Scholar
  32. 32.
    Lera, D., Sergeyev, Y.D.: Lipschitz and Hölder global optimization using space-filling curves. Appl. Numer. Math. 60(1–2), 115–129 (2010). doi: 10.1016/j.apnum.2009.10.004 CrossRefGoogle Scholar
  33. 33.
    Lera, D., Sergeyev, Y.D.: Acceleration of univariate global optimization algorithms working with Lipschitz functions and Lipschitz first derivatives. SIAM J. Optim. 23(1), 508–529 (2013). doi: 10.1137/110859129 CrossRefGoogle Scholar
  34. 34.
    Liu, Q.: Linear scaling and the direct algorithm. J. Global Optim. 56, 1233–1245 (2013). doi: 10.1007/s10898-012-9952-x CrossRefGoogle Scholar
  35. 35.
    Liu, Q., Cheng, W.: A modified direct algorithm with bilevel partition. J. Global Optim. 1–17 (2013). doi: 10.1007/s10898-013-0119-1
  36. 36.
    Liuzzi, G., Lucidi, S., Piccialli, V.: A direct-based approach exploiting local minimizations for the solution for large-scale global optimization problems. Comput. Optim. Appl. 45(2), 353–375 (2010). doi: 10.1007/s10589-008-9217-2 CrossRefGoogle Scholar
  37. 37.
    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). doi: 10.1007/s10589-008-9217-2 CrossRefGoogle Scholar
  38. 38.
    Moré, J.J., Wild, S.M.: Benchmarking derivative-free optimization algorithms. SIAM J. Optim. 20(1), 172–191 (2009). doi: 10.1137/080724083 CrossRefGoogle Scholar
  39. 39.
    Pardalos, P.M., Romeijn, H.E. (eds.): Handbook of Global Optimization, vol. 2. Kluwer Academic Publishers, Dordrecht (2002)Google Scholar
  40. 40.
    Paulavičius, R., Žilinskas, J.: Analysis of different norms and corresponding Lipschitz constants for global optimization in multidimensional case. Inf. Technol. Control 36(4), 383–387 (2007)Google Scholar
  41. 41.
    Paulavičius, R., Žilinskas, J.: Influence of Lipschitz bounds on the speed of global optimization. Technol. Econ. Dev. Econ. 18(1), 54–66 (2012). doi: 10.3846/20294913.2012.661170 CrossRefGoogle Scholar
  42. 42.
    Paulavičius, R., Žilinskas, J.: Simplicial Lipschitz optimization without the Lipschitz constant. J. Global Optim. 59(1), 23–40 (2014). doi: 10.1007/s10898-013-0089-3 Google Scholar
  43. 43.
    Paulavičius, R., Žilinskas, J.: Advantages of simplicial partitioning for Lipschitz optimization problems with linear constraints. Optim. Lett. (2014) SubmittedGoogle Scholar
  44. 44.
    Paulavičius, R., Žilinskas, J.: Simplicial Global Optimization. SpringerBriefs in Optimization. Springer, New York (2014). doi: 10.1007/978-1-4614-9093-7 CrossRefGoogle Scholar
  45. 45.
    Paulavičius, R., Žilinskas, J., Grothey, A.: Investigation of selection strategies in branch and bound algorithm with simplicial partitions and combination of Lipschitz bounds. Optim. Lett. 4(2), 173–183 (2010). doi: 10.1007/s11590-009-0156-3 CrossRefGoogle Scholar
  46. 46.
    Pintér, J.D.: Global Optimization in Action (Continuous and Lipschitz Optimization: Algorithms, Implementations and Applications). Kluwer Academic Publishers, Dordrecht (1996)CrossRefGoogle Scholar
  47. 47.
    Sergeyev, Y.D.: An information global optimization algorithm with local tuning. SIAM J. Optim. 5(4), 858–870 (1995). doi: 10.1137/0805041 CrossRefGoogle Scholar
  48. 48.
    Sergeyev, Y.D.: Global one-dimensional optimization using smooth auxiliary functions. Math. Program. 81(1), 127–146 (1998). doi: 10.1007/BF01584848 CrossRefGoogle Scholar
  49. 49.
    Sergeyev, Y.D.: An efficient strategy for adaptive partition of \(N\)-dimensional intervals in the framework of diagonal algorithms. J. Optim. Theory Appl. 107(1), 145–168 (2000). doi:  10.1023/A:1004613001755 CrossRefGoogle Scholar
  50. 50.
    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). doi: 10.1137/040621132 CrossRefGoogle Scholar
  51. 51.
    Sergeyev, Y.D., Kvasov, D.E.: Diagonal Global Optimization Methods. FizMatLit, Moscow (2008). In RussianGoogle Scholar
  52. 52.
    Sergeyev, Y.D., Kvasov, D.E.: Lipschitz global optimization. In: Cochran, J.J., Cox, L.A., Keskinocak, P., Kharoufeh, J.P., Smith, J.C. (eds.) Wiley Encyclopedia of Operations Research and Management Science (in 8 volumes), vol. 4, pp. 2812–2828. Wiley, New York (2011)Google Scholar
  53. 53.
    Sergeyev, Y.D., Strongin, R.G., Lera, D.: Introduction to Global Optimization Exploiting Space-Filling Curves. SpringerBriefs in Optimization. Springer, New York (2013). doi: 10.1007/978-1-4614-8042-6 CrossRefGoogle Scholar
  54. 54.
    Strongin, R.G., Sergeyev, Y.D.: Global Optimization with Non-Convex Constraints: Sequential and Parallel Algorithms. Kluwer Academic Publishers, Dordrecht (2000)CrossRefGoogle Scholar
  55. 55.
    Todt, M.J.: The Computation of Fixed Points and Applications. Lecture Notes in Economics and Mathematical Systems, vol. 24. Springer, Berlin (1976)Google Scholar
  56. 56.
    Zhigljavsky, A., Žilinskas, A.: Stochastic Global Optimization. Springer, New York (2008)Google Scholar
  57. 57.
    Žilinskas, A.: On strong homogeneity of two global optimization algorithms based on statistical models of multimodal objective functions. Appl. Math. Comput. 218(16), 8131–8136 (2012). doi: 10.1016/j.amc.2011.07.051 CrossRefGoogle Scholar
  58. 58.
    Žilinskas, A., Žilinskas, J.: Global optimization based on a statistical model and simplicial partitioning. Comput. Math. Appl. 44(7), 957–967 (2002). doi: 10.1016/S0898-1221(02)00206-7 CrossRefGoogle Scholar
  59. 59.
    Žilinskas, A., Žilinskas, J.: A hybrid global optimization algorithm for non-linear least squares regression. J. Global Optim. 56(2), 265–277 (2013). doi: 10.1007/s10898-011-9840-9 CrossRefGoogle Scholar
  60. 60.
    Žilinskas, J.: Branch and bound with simplicial partitions for global optimization. Math. Model. Anal. 13(1), 145–159 (2008). doi: 10.3846/1392-6292.2008.13.145-159 CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Remigijus Paulavičius
    • 1
  • Yaroslav D. Sergeyev
    • 2
    • 3
    Email author
  • Dmitri E. Kvasov
    • 2
    • 3
  • Julius Žilinskas
    • 1
  1. 1.Institute of Mathematics and InformaticsVilnius UniversityVilniusLithuania
  2. 2.Dipartimento di Ingegneria Informatica, Modellistica, Elettronica e SistemisticaUniversità della CalabriaRendeItaly
  3. 3.Software DepartmentN. I. Lobachevsky State UniversityNizhniy NovgorodRussia

Personalised recommendations