Skip to main content
SpringerLink
Log in

On a Global Optimization Algorithm for Bivariate Smooth Functions

  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

Abstract

The problem of approximating the global minimum of a function of two variables is considered. A method is proposed rooted in the statistical approach to global optimization. The proposed algorithm partitions the feasible region using a Delaunay triangulation. Only the objective function values are required by the optimization algorithm. The asymptotic convergence rate is analyzed for a class of smooth functions. Numerical examples are provided.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

References

  1. Mockus, J.: Bayesian Approach to Global Optimization. Kluwer Academic Publishers, Dordrecht (1988)

  2. Törn, A., Žilinskas, A.: Global optimization. Lect. Notes Comput. Sci. 350, 1–255 (1989)

    Article  Google Scholar 

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

  4. Zhigljavsy, A., Žilinskas, A.: Stochastic Global Optimization. Springer, New York (2008)

  5. Calvin, J.M., Žilinskas, A.: A one-dimensional P-algorithm with convergence rate \(O(n^{-3+\delta })\) for smooth functions. J. Optim. Theory Appl. 106, 297–307 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  6. Calvin, J.M., Chen, Y., Žilinskas, A.: An adaptive univariate global optimization algorithm and its convergence rate for twice continuously differentiable functions. J. Optim. Theory Appl. 155, 628–636 (2012)

    Google Scholar 

  7. Horst, R., Pardalos, P., Thoai, N.: Introduction to Global Optimization. Kluwer Academic Publishers, Dordrecht (1995)

  8. Scholz, D.: Deterministic Global Optimization: Geometric Branch-and-Bound Methods and their Applications. Springer, New York (2012)

  9. Sergeyev, Y., Kvasov, D.: Diagonal Global Optimization Methods (in Russian). Fizmatlit, Moscow (2008)

  10. Pinter, J.: Global Optimization in Action. Kluwer Academic Publisher, Dordrecht (1996)

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

    Article  MathSciNet  MATH  Google Scholar 

  12. Baritompa, W.: Customizing methods for global optimization: a geometric viewpoint. J. Glob. Opt. 3(2), 193–212 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  13. Clausen, J., Žilinskas, A.: Global optimization by means of branch and bound with simplex based covering. Comput. Math. Appl. 44, 943–995 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  14. Huyer, W., Neumaier, A.: Global optimization by multi-level coordinate search. J. Glob. Opt. 14, 331–355 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  15. Kvasov, D., Sergeyev, Y.: Lipschitz gradients for global optimization in a one-point-based partitioning scheme. J. Comput. Appl. Math. 236(16), 4042–4054 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  16. Liuzzi, G., Lucidi, S., Picciali, V.: A partition-based global optimization algorithm. J. Glob. Opt. 48(1), 113–128 (2010)

    Article  MATH  Google Scholar 

  17. Sergeyev, Y., Strongin, R., Lera, D.: Introduction to Global Optimization Exploiting Space-Filling Curves. Springer, New York (2013)

  18. Wood, G.: Bisection global optimization methods. In: Floudas, C., Pardalos, P. (eds.) Encyclopedia of Optimization, pp. 186–189. Kluwer Academic Publisher, Dordrecht (2001)

    Chapter  Google Scholar 

  19. Žilinskas, A., Žilinskas, J.: Global optimization based on a statistical model and simplicial partitioning. Comput. Math. Appl. 44, 957–967 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  20. Žilinskas, A., Žilinskas, J.: A global optimization method based on the reduced simplicial statistical model. Math. Model. Anal. 16, 451–460 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  21. Paulavičius, R., Žilinskas, J., Grothey, A.: Investigation of selection strategies in branch and bound algorithm with simplicial partition and combination of Lipschitz bounds. Optim. Lett. 4, 173–183 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  22. Hendrix, E., Casado, L., Garcia, I.: The semi-continuous quadratic mixture design problem: description and branch-and-bound approach. Eur. J. Oper. Res. 191(3), 803–815 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  23. Wu, Y., Ozdamar, L., Kumar, A.: Triopt: a triangulation-based partitioning algorithm for gloabal optimization. J. Comput. Appl. Math. 177, 35–53 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  24. Fishburn, P.: Utility Theory for Decision Making. Wiley, New York (1970)

  25. Žilinskas, A.: Axiomatic approach to statistical models and their use in multimodal optimizatio theory. Math. Program. 22, 104–116 (1982)

    Article  MATH  Google Scholar 

  26. Kushner, H.: A versatile stochastic model of a function of unknown and time-varying form. J. Math. Anal. Appl. 5, 150–167 (1962)

    Article  MathSciNet  MATH  Google Scholar 

  27. Žilinskas, A.: Axiomatic characterization of a global optimization algorithm and investigation of its search strategies. Oper. Res. Lett. 4, 35–39 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  28. Lawson, C.: Software for C1 surface interpolation. In: Rice, J.R. (ed.) Mathematical Software III, vol. 1, pp. 161–194. Academic Press, New York (1977)

    Chapter  Google Scholar 

  29. Hjelle, Ø., Dælen, M.: Triangulations and Applications. Springer-Verlag, New York (2006)

  30. Shewchuck, J.R.: Delaunay refinement algorithms for triangular mesh generation. Comput. Geom. 22, 21–31 (2002)

    Article  MathSciNet  Google Scholar 

  31. Cgal, Computational Geometry Algorithms Library. URL http://www.cgal.org. Accessed 1 Oct 2013

  32. Waldron, S.: Sharp error estimates for multivariate positive linear operators which reproduce the linear polynomials. In: Chui, C.K., Schumaker, L.L. (eds.) Approximation Theory IX, vol. 1, pp. 339–346. Vanderbilt University Press, Nashville (1998)

  33. Rastrigin, L.: Statistical Models of Search (in Russian). Nauka (1968).

  34. Branin, F.: Widely convergent method for finding multiple solutions of simultaneous nonlinear equations. IBM J. Res. Dev. 16(5), 504–522 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  35. Dixon, L.C.W., Szegö, G.P. (eds.): Towards Global Optimization II. North Holland, New York (1978)

  36. DIRECT - A Global Optimization Algorithm. URL http://www4.ncsu.edu/ctk/Finkel.Direct. Accessed 1 Oct 2013

  37. Hansen, P., Jaumard, B.: Lipshitz optimization. In: Horst, R., Pardalos, P. (eds.) Handbook of Global Optimization, pp. 407–493. Kluwer Academic Publisher, Dodrecht (1995)

Download references

Acknowledgments

The work of J. Calvin was supported by the National Science Foundation under Grant No. CMMI-0825381, and the work of A. Žilinskas was supported by the Research Council of Lithuania under Grant No. MIP-063/2012. The remarks of unknown referees were very helpful to improve the readability of the paper.

Author information

Authors and Affiliations

  1. Department of Computer Science, New Jersey Institute of Technology, Newark, NJ, 07102-1982, USA

    James M. Calvin

  2. Institute of Mathematics and Informatics, Vilnius University, Akademijos str. 4, Vilnius, 08663, Lithuania

    Antanas Žilinskas

Authors
  1. James M. Calvin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Antanas Žilinskas
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Antanas Žilinskas.

Additional information

Communicated by Panos M. Pardalos.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Calvin, J.M., Žilinskas, A. On a Global Optimization Algorithm for Bivariate Smooth Functions. J Optim Theory Appl 163, 528–547 (2014). https://doi.org/10.1007/s10957-014-0531-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-014-0531-9

Keywords

Mathematics Subject Classification

Search

Navigation