On Two Approaches to Constructing Optimal Algorithms for Multi-objective Optimization

  • Antanas Žilinskas
Part of the Communications in Computer and Information Science book series (CCIS, volume 403)


Multi-objective optimization problems with expensive, black box objectives are difficult to tackle. For such type of problems in the single objective case the algorithms, which are in some sense optimal, have proved well suitable. Two concepts of optimality substantiate the construction of algorithms: worst case optimality and average case optimality. In the present paper the extension of these concepts to the multi-objective optimization is discussed. Two algorithms representing both concepts are implemented and experimentally compared.


Multi-objective optimization global optimization optimal algorithms statistical models 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Arora, S., Barak, B.: Computational Complexity a Modern Approach. Cambridge University Press (2009)Google Scholar
  2. 2.
    Calvin, J., Žilinskas, A.: A one-dimensional P-algorithm with convergence rate O(n − 3 + δ). J. Optimization Theory and Applications 106, 297–307 (2000)CrossRefzbMATHGoogle Scholar
  3. 3.
    Deb, K.: Multi-Objective Optimization Using Evolutionary Algorithms. J. Wiley, Chichester (2009)zbMATHGoogle Scholar
  4. 4.
    Fishburn P. (1970). Utility Theory for Decision Making. J.Wiley, Chichester.zbMATHGoogle Scholar
  5. 5.
    Fonseca, C., Fleming, P.: On the performance assessment and comparison of multiobjective optimizers. In: Ebeling, W., Rechenberg, I., Voigt, H.-M., Schwefel, H.-P. (eds.) PPSN 1996. LNCS, vol. 1141, pp. 584–593. Springer, Heidelberg (1996)CrossRefGoogle Scholar
  6. 6.
    Horst, R., Pardalos, P., Thoai, N.: Introduction to Global Optimization. KAP, Boston (2000)Google Scholar
  7. 7.
    Jančauskas, V., Mackutė-Varoneckienė, A., Varoneckas, A., Žilinskas, A.: Multi-objective optimization aided visualization of graphs related to business process management. Comm. in Comp. and Inform. Sci. 319, 87–100 (2012)CrossRefGoogle Scholar
  8. 8.
    Kushner, H.: A versatile stochastic model of a function of unknown and time-varying form. J. Math. Anal. and Appl. 5, 150–167 (1962)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Miettinen, K.: l Nonlinear multiobjective optimization. KAP, Boston (1999)Google Scholar
  10. 10.
    Mockus, J.: Bayesian approach to global optimization. KAP, Boston (1988)Google Scholar
  11. 11.
    Nakayama, H., Yun, Y., Yoon, M.: Sequential Approximate Multiobjective Optimization Using Computational Intelligence. Springer, Berlin (2009)zbMATHGoogle Scholar
  12. 12.
    Pijavskii, S.: An algorithm for finding the absolute extremum of a function. USSR Computational Mathematics and Mathematical Physics 12, 57–67 (1972)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Shubert, B.: A sequential method seeking the global maximum of a function. SIAM J. Numer. Anal. 9, 379–388 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Strongin, R., Sergeyev, Y.: Global Optimization with Non-convex Constraints: Sequential and Parallel Algorithms. KAP, Boston (2000)CrossRefGoogle Scholar
  15. 15.
    Törn, A., Žilinskas, A.: Global Optimization. LNCS, vol. 350, pp. 1–225. Springer, Heidelberg (1989)zbMATHGoogle Scholar
  16. 16.
    Sukharev, A.: On optimal strategies of search for an extremum. USSR Comput. Math. and Math. Physics 11, 910–924 (1971) (in Russian)Google Scholar
  17. 17.
    Sukharev, A.: Best strategies of sequential search for an extremum. USSR Comput. Math. and Math. Physics 12, 35–50 (1972) (in Russian)Google Scholar
  18. 18.
    Žilinskas, A.: Optimization of one-dimensional multimodal functions, Algorithm AS-133. Journal of Royal Statistical Society, ser. C 23, 367–385 (1978)Google Scholar
  19. 19.
    Žilinskas, A.: Axiomatic approach to statistical models and their use in multimodal optimizatio theory. Mathematical Programming 22, 104–116 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Žilinskas, A.: Axiomatic characterization of a global optimization algorithm and investigation of its search strategy. Operat. Res. Letters 4, 35–39 (1985)CrossRefzbMATHGoogle Scholar
  21. 21.
    Žilinskas, A.: A statistical model-based algorithm for black-box multi-objective optimization. International Journal of Systems Science (2012) (Published on Internet July 4, 2012), doi:10.1080/00207721.2012.702244Google Scholar
  22. 22.
    Žilinskas, A.: On the worst-case optimal multi-objective global optimization. Optimization Letters (2012) (Published on Internet September 14, 2012), doi:10.1007/s11590-012-0547-8Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Antanas Žilinskas
    • 1
  1. 1.Institute of Mathematics and InformaticsVilnius UniversityVilniusLithuania

Personalised recommendations