Skip to main content
SpringerLink
Log in

On efficiency of a single variable bi-objective optimization algorithm

  • Original Paper
  • Published:
Optimization Letters Aims and scope Submit manuscript

Abstract

We consider the problem of approximating the Pareto front in the bi-objective optimization problem with twice continuously-differentiable functions. An algorithm is described that is efficient in the sense that the number of function evaluations that are not on the true Pareto front grows as the square of the logarithm of the number of function evaluations.

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. Bökler, F., Ehrgott, M., Morris, C., Mutzel, P.: Output-sensitive complexity of multiobjective combinatorial optimization. 1–21 (2016) arXiv:1610.07204v1

  2. Calvin, J., Gimbutienė, G., Phillips, W.O., Žilinskas, A.: On convergence rate of a rectangular partition based global optimization algorithm. J. Glob. Optim. 71, 165–191 (2018)

    Article  MathSciNet  Google Scholar 

  3. Calvin, J.M., Chen, Y., Z̆ilinskas, A.: An adaptive univariate global optimization algorithm and its convergence rate for twice continuously differentiable functions. J. Optim. Theory Appl. 155(2), 628–636 (2012)

    Article  MathSciNet  Google Scholar 

  4. Calvin, J.M., Z̆ilinskas, A.: On convergence of the P-algorithm for one dimensional global optimization of smooth functions. J. Optim. Theory Appl. 102, 479–495 (1999)

    Article  MathSciNet  Google Scholar 

  5. Calvin, J.M.: An adaptive univariate global optimization algorithm and its convergence rate under the Wiener measure. Informatica 22, 471–488 (2010)

    MathSciNet  MATH  Google Scholar 

  6. Calvin, J.M., Hefter, M., Herzwurm, A.: Adaptive approximation of the minimum of brownian motion. J. Complex. 39(C), 17–37 (2017)

    Article  MathSciNet  Google Scholar 

  7. Deb, K.: Multi-objective Optimization Using Evolutionary Algorithms. Wiley, Hoboken (2009)

    MATH  Google Scholar 

  8. Fliege, J., Vaz, A.I.F., Vicente, L.N.: Complexity of Gradient Descent for Multiobjective Optimization. https://www.mat.uc.pt/~lnv/papers/wcc-moo.pdf (2018). Accessed 11 Feb 2019

    Article  MathSciNet  Google Scholar 

  9. Laumanns, M., Thiele, L., Zitzler, E.: An efficient, adaptive parameter variation scheme for metaheuristics based on the epsilon-constraint method. Eur. J. Oper. Res. 169, 932–942 (2006)

    Article  MathSciNet  Google Scholar 

  10. Novak, E., Woźniakowski, H.: Tractability of Multivariate Problems Volume II: Standard Information for Functionals. European Mathematical Society, Berlin (2010)

    Book  Google Scholar 

  11. Pardalos, P.M., Žilinskas, A., Žilinskas, J.: Non-convex Multi-objective Optimization. Springer, Berlin (2017)

    Book  Google Scholar 

  12. Serafini, P.: Some considerations about computational complexity for multi objective combinatorial problems. In: Jahn, J., Krabs, W. (eds.) Recent Advances and Historical Development of Vector Optimization, pp. 222–232. Kluwer Academic Publishers, Dodrecht (1987)

    Chapter  Google Scholar 

  13. Strongin, R., Sergeyev, Y.: Global Optimization with Non-convex Constrints. Kluwer Academic Publishers, Dodrecht (2000)

    Book  Google Scholar 

  14. Žilinskas, A.: On the worst-case optimal multi-objective global optimization. Optim. Lett. 7, 1921–1928 (2013)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The work of the first author was supported by the National Science Foundation under Grant No. CMMI-1562466. The work of second author was supported by the Research Council of Lithuania under Grant No. P-MIP-17-61. We thank the associate editor for his valuable remarks enabling us to improve the presentation of our results.

Author information

Authors and Affiliations

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

    James M. Calvin

  2. Institute of Data Science and Digital Technologies, Vilnius University, Akademijos 4, 08663, Vilnius, 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 James M. Calvin.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Calvin, J.M., Žilinskas, A. On efficiency of a single variable bi-objective optimization algorithm. Optim Lett 14, 259–267 (2020). https://doi.org/10.1007/s11590-019-01471-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11590-019-01471-4

Keywords

Search

Navigation