Globally-biased Disimpl algorithm for expensive global optimization

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.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4

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)

  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

    Article  Google 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

    Article  Google 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

    Article  Google 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

    Article  Google 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

    Article  Google 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

    Article  Google 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

    Article  Google 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

    Article  Google 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

    Article  Google Scholar 

  12. 12.

    Finkel, D.E.: Global Optimization with the Direct Algorithm. Ph.D. thesis, North Carolina State University (2005)

  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

    Article  Google 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)

  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

    Article  Google 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

    Article  Google 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 Russian

    Google 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 Russian

    Google 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

    Article  Google 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 Russian

  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

    Article  Google 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)

    Google 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)

    Google 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

    Article  Google 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

    Article  Google 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

    Article  Google 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

    Article  Google 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

    Article  Google 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

    Article  Google 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

    Article  Google 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

    Article  Google 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

    Article  Google 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

    Article  Google 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

    Article  Google 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

    Article  Google 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

    Article  Google 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) Submitted

  44. 44.

    Paulavičius, R., Žilinskas, J.: Simplicial Global Optimization. SpringerBriefs in Optimization. Springer, New York (2014). doi:10.1007/978-1-4614-9093-7

    Google 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

    Article  Google Scholar 

  46. 46.

    Pintér, J.D.: Global Optimization in Action (Continuous and Lipschitz Optimization: Algorithms, Implementations and Applications). Kluwer Academic Publishers, Dordrecht (1996)

    Google 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

    Article  Google 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

    Article  Google 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

    Article  Google 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

    Article  Google Scholar 

  51. 51.

    Sergeyev, Y.D., Kvasov, D.E.: Diagonal Global Optimization Methods. FizMatLit, Moscow (2008). In Russian

    Google 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

    Google Scholar 

  54. 54.

    Strongin, R.G., Sergeyev, Y.D.: Global Optimization with Non-Convex Constraints: Sequential and Parallel Algorithms. Kluwer Academic Publishers, Dordrecht (2000)

    Google 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

    Article  Google 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

    Article  Google 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

    Article  Google 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

    Article  Google Scholar 

Download references

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.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Yaroslav D. Sergeyev.

Additional information

Dedicated to Panos M. Pardalos on the occasion of his 60th birthday.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Paulavičius, R., Sergeyev, Y.D., Kvasov, D.E. et al. Globally-biased Disimpl algorithm for expensive global optimization. J Glob Optim 59, 545–567 (2014). https://doi.org/10.1007/s10898-014-0180-4

Download citation

Keywords

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