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Optimization Letters

, Volume 7, Issue 8, pp 1921–1928 | Cite as

On the worst-case optimal multi-objective global optimization

  • Antanas ŽilinskasEmail author
Original Paper

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.

Keywords

Multi-objective optimization Optimal algorithms Global optimization Lipshitz optimization 

Notes

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.

References

  1. 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)Google Scholar
  2. 2.
    Chinchuluun, A., Pardalos, P.M.: A survey of recent developments in multiobjective optimization. Ann. Oper. Res. 154(1), 29–50 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Deb, K.: Multi-Objective Optimization Using Evolutionary Algorithms. Wiley, New York (2009)zbMATHGoogle Scholar
  4. 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)Google Scholar
  5. 5.
    Horst, R., Pardalos, P., Thoai, N.: Introduction to Global Optimization. Kluwer, Dordrecht (2000)CrossRefzbMATHGoogle Scholar
  6. 6.
    Miettinen, K.: Nonlinear Multiobjective Optimization. Springer, Berlin (1999)zbMATHGoogle Scholar
  7. 7.
    Nakayama, H.: Sequential Approximate Multiobjective Optimization Using Computational Intelligence. Springer, Berlin (2009)zbMATHGoogle Scholar
  8. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Sayin, S.: Measuring the quality of discrete representation of efficient sets in multiple objective mathematical programming. Math. Program. 87 A, 543–560 (2000)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Sukharev, A.: On optimal strategies of search for an extremum (in Russian). USSR Comput. Math. Math. Phys. 11(4), 910–924 (1971)CrossRefGoogle Scholar
  11. 11.
    Sukharev, A.: Best strategies of sequential search for an extremum (in Russian). USSR Comput. Math. Math. Phys. 12(1), 35–50 (1972)CrossRefGoogle Scholar
  12. 12.
    Sukharev, A.: A sequentially optimal algorithm for numerical integration. J. Optim. Theory Appl. 28(3), 363–373 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Törn, A., Žilinskas, A.: Global optimization. Lect. Notes Comput. Sci. 350, 1–252 (1989)CrossRefGoogle Scholar
  14. 14.
    Traub, J.F., Wasilkowski, G.W., Wozniakowski, H.: Information, Uncertainty, Complexity. Addison-Wesley, Reading (1983)zbMATHGoogle Scholar
  15. 15.
    Zhigljavsky, A., Žilinskas, A.: Stochastic Global Optimization. Springer, Berlin (2008)Google Scholar
  16. 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. 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)CrossRefGoogle Scholar
  18. 18.
    Zopounidis, C., Pardalos, P.: Handbook of Multicriteria Analysis. Springer, Berlin (2010)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  1. 1.Institute of Mathematics and InformaticsVilniusLithuania

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