Skip to main content
Log in

On the worst-case optimal multi-objective global optimization

Optimization Letters Aims and scope Submit manuscript

Abstract

Multi-objective optimization problem with Lipshitz objective functions is considered. It is shown that the worst-case optimal passive algorithm can be reduced to the computation of centers of balls producing the optimal cover of a feasible region, where the balls are of equal minimum radius. It is also shown, that in the worst-case, adaptivity does not improve the guaranteed accuracy achievable by the passive algorithm.

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

Access this article

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

Instant access to the full article PDF.

Institutional subscriptions

References

  1. Branke, J., Deb, K., Miettinen, K., Słowiński, R., (eds.): Multiobjective Optimization: Interactive and Evolutionary Approaches; Dagstuhl Seminar on Practical Approaches to Multi-Objective Optimization, Schloss Dagstuhl, Dec 10–15, 2006. Lecture Notes in Computer Science, vol. 5252. Springer, Berlin (2008)

  2. Chinchuluun, A., Pardalos, P.M.: A survey of recent developments in multiobjective optimization. Ann. Oper. Res. 154(1), 29–50 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  3. Deb, K.: Multi-Objective Optimization Using Evolutionary Algorithms. Wiley, New York (2009)

    MATH  Google Scholar 

  4. Fonseca, C., Fleming, P.: On the performance assessment and comparison of stochastic multiobjective optimizers. In: Ebeling, W., Rechenberg, I., Schwefel, H.-P., Voigt, H.-M. (eds.) Parallel Problem Solving from Nature, Berlin, September 22–26. Lecture notes in Computer Science, vol. 1141, pp. 584–593. Springer, Berlin (1996)

  5. Horst, R., Pardalos, P., Thoai, N.: Introduction to Global Optimization. Kluwer, Dordrecht (2000)

    Book  MATH  Google Scholar 

  6. Miettinen, K.: Nonlinear Multiobjective Optimization. Springer, Berlin (1999)

    MATH  Google Scholar 

  7. Nakayama, H.: Sequential Approximate Multiobjective Optimization Using Computational Intelligence. Springer, Berlin (2009)

    MATH  Google Scholar 

  8. Pardalos, P., Steponavice, I., Žilinskas, A.: Pareto set approximation by the method of adjustable weights and successive lexicographic goal programming. Optim. Lett. 6, 665–678 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  9. Sayin, S.: Measuring the quality of discrete representation of efficient sets in multiple objective mathematical programming. Math. Program. 87 A, 543–560 (2000)

    Article  MathSciNet  Google Scholar 

  10. Sukharev, A.: On optimal strategies of search for an extremum (in Russian). USSR Comput. Math. Math. Phys. 11(4), 910–924 (1971)

    Article  Google Scholar 

  11. Sukharev, A.: Best strategies of sequential search for an extremum (in Russian). USSR Comput. Math. Math. Phys. 12(1), 35–50 (1972)

    Article  Google Scholar 

  12. Sukharev, A.: A sequentially optimal algorithm for numerical integration. J. Optim. Theory Appl. 28(3), 363–373 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  13. Törn, A., Žilinskas, A.: Global optimization. Lect. Notes Comput. Sci. 350, 1–252 (1989)

    Article  Google Scholar 

  14. Traub, J.F., Wasilkowski, G.W., Wozniakowski, H.: Information, Uncertainty, Complexity. Addison-Wesley, Reading (1983)

    MATH  Google Scholar 

  15. Zhigljavsky, A., Žilinskas, A.: Stochastic Global Optimization. Springer, Berlin (2008)

    Google Scholar 

  16. Žilinskas, A.: A statistical model-based algorithm for black-box multi-objective optimization. Int. J. Syst. Sci. (2012). Published on Internet 4 July, 2012. doi:10.1080/00207721.2012.702244

  17. Zitzler, E., Thiele, L., Laummanns, M., Fonseca, C.M., da Fonseca, Grunert: Performance assessment of multiobjective optimizers: an analysis and review. IEEE Trans. Evol. Comput. 3(4), 257–271 (2003)

    Article  Google Scholar 

  18. Zopounidis, C., Pardalos, P.: Handbook of Multicriteria Analysis. Springer, Berlin (2010)

    Book  MATH  Google Scholar 

Download references

Acknowledgments

This research is supported by the Research Council of Lithuania under Grant No. MIP-063/2012. The constructive remarks of two referees facilitated a significant improvement of the presentation of results.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Antanas Žilinskas.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Žilinskas, A. On the worst-case optimal multi-objective global optimization. Optim Lett 7, 1921–1928 (2013). https://doi.org/10.1007/s11590-012-0547-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11590-012-0547-8

Keywords

Navigation