Skip to main content
SpringerLink
Log in

Worst-Case Complexity Bounds of Directional Direct-Search Methods for Multiobjective Optimization

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

Abstract

Direct Multisearch is a well-established class of algorithms, suited for multiobjective derivative-free optimization. In this work, we analyze the worst-case complexity of this class of methods in its most general formulation for unconstrained optimization. Considering nonconvex smooth functions, we show that to drive a given criticality measure below a specific positive threshold, Direct Multisearch takes at most a number of iterations proportional to the square of the inverse of the threshold, raised to the number of components of the objective function. This number is also proportional to the size of the set of linked sequences between the first unsuccessful iteration and the iteration immediately before the one where the criticality condition is satisfied. We then focus on a particular instance of Direct Multisearch, which considers a more strict criterion for accepting new nondominated points. In this case, we can establish a better worst-case complexity bound, simply proportional to the square of the inverse of the threshold, for driving the same criticality measure below the considered threshold.

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

Similar content being viewed by others

References

  1. Marler, R.T., Arora, J.S.: Survey of multi-objective optimization methods for engineering. Struct. Multidiscip. Optim. 26, 369–395 (2004)

    Article  MathSciNet  Google Scholar 

  2. Ehrgott, M.: Multicriteria Optimization. Springer, Heidelberg (2005)

    MATH  Google Scholar 

  3. Custódio, A.L., Emmerich, M., Madeira, J.F.A.: Recent developments in derivative-free multiobjective optimization. Comput. Technol. Rev. 5, 1–30 (2012)

    Article  Google Scholar 

  4. Kolda, T.G., Lewis, R.M., Torczon, V.: Optimization by direct search: new perspectives on some classical and modern methods. SIAM Rev. 45, 385–482 (2003)

    Article  MathSciNet  Google Scholar 

  5. Conn, A.R., Scheinberg, K., Vicente, L.N.: Introduction to Derivative-Free Optimization. MPS-SIAM Series on Optimization. SIAM, Philadelphia (2009)

    Book  Google Scholar 

  6. Audet, C., Hare, W.: Derivative-Free and Blackbox Optimization. Springer Series in Operations Research and Financial Engineering. Springer, Cham (2017)

    Book  Google Scholar 

  7. Custódio, A.L., Madeira, J.F.A., Vaz, A.I.F., Vicente, L.N.: Direct multisearch for multiobjective optimization. SIAM J. Optim. 21, 1109–1140 (2011)

    Article  MathSciNet  Google Scholar 

  8. Cartis, C., Gould, N.I.M., Toint, Ph.L.: On the oracle complexity of first-order and derivative-free algorithms for smooth nonconvex minimization. SIAM J. Optim. 22, 66–86 (2012)

  9. Dodangeh, M., Vicente, L.N.: Worst case complexity of direct search under convexity. Math. Program. 155, 307–332 (2016)

    Article  MathSciNet  Google Scholar 

  10. Fliege, J., Vaz, A.I.F., Vicente, L.N.: Complexity of gradient descent for multiobjective optimization. Optim. Methods Softw. 34, 949–959 (2019)

    Article  MathSciNet  Google Scholar 

  11. Garmanjani, R., Vicente, L.N.: Smoothing and worst-case complexity for direct-search methods in nonsmooth optimization. IMA J. Numer. Anal. 33, 1008–1028 (2013)

    Article  MathSciNet  Google Scholar 

  12. Grapiglia, G.N., Yuan, J., Yuan, Y.: On the convergence and worst-case complexity of trust-region and regularization methods for unconstrained optimization. Math. Program. 152, 491–520 (2015)

    Article  MathSciNet  Google Scholar 

  13. Gratton, S., Royer, C.W., Vicente, L.N., Zhang, Z.: Direct search based on probabilistic descent. SIAM J. Optim. 25, 1515–1541 (2015)

    Article  MathSciNet  Google Scholar 

  14. Konečný, J., Richtárik, P.: Simple Complexity Analysis of Simplified Direct Search. Technical reports, arXiv:1410.0390 (2014)

  15. Nesterov, Y.: Introductory Lectures on Convex Optimization. Applied Optimization. Kluwer Academic Publishers, Boston (2004)

    Book  Google Scholar 

  16. Vicente, L.N.: Worst case complexity of direct search. EURO J. Comput. Optim. 1, 143–153 (2013)

    Article  Google Scholar 

  17. Gratton, S., Sartenaer, A., Toint, Ph.L.: Recursive trust-region methods for multiscale nonlinear optimization. SIAM J. Optim. 19, 414–444 (2008)

  18. Cartis, C., Sampaio, Ph.R., Toint, Ph.L.: Worst-case evaluation complexity of non-monotone gradient-related algorithms for unconstrained optimization. Optimization 64, 1349–1361 (2015)

  19. Nesterov, Y., Polyak, B.T.: Cubic regularization of Newton method and its global performance. Math. Program. 108, 177–205 (2006)

    Article  MathSciNet  Google Scholar 

  20. Cartis, C., Gould, N.I.M., Toint, Ph.L.: Adaptive cubic regularisation methods for unconstrained optimization. Part II: worst-case function- and derivative-evaluation complexity. Math. Program. 130, 295–319 (2011)

  21. Bergou, E.H., Gorbunov, E., Richtárik, P.: Stochastic three points method for unconstrained smooth optimization. SIAM J. Optim. 30, 2726–2749 (2020)

    Article  MathSciNet  Google Scholar 

  22. Calderón, L., Diniz-Ehrhardt, M.A., Martínez, J.M.: On high-order model regularization for multiobjective optimization. Optim. Methods Softw. (2020). https://doi.org/10.1080/10556788.2020.1719408

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

    Article  MathSciNet  Google Scholar 

  24. Calvin, J.M., Žilinskas, A.: On efficiency of a single variable bi-objective optimization algorithm. Optim. Lett. 14, 259–267 (2020)

    Article  MathSciNet  Google Scholar 

  25. Villacorta, K.D.V., Oliveira, P.R., Soubeyran, A.: A trust-region method for unconstrained multiobjective problems with applications in satisficing processes. J. Optim. Theory Appl. 160, 865–889 (2014)

    Article  MathSciNet  Google Scholar 

  26. Fliege, J., Svaiter, B.F.: Steepest descent methods for multicriteria optimization. Math. Methods Oper. Res. 51, 479–494 (2000)

    Article  MathSciNet  Google Scholar 

  27. Zitzler, E.: Evolutionary Algorithms for Multiobjective Optimization: Methods and Applications. Ph.D. Thesis, Swiss Federal Institute of Technology Zurich, Switzerland (1999)

  28. Thomann, J., Eichfelder, G.: A trust-region algorithm for heteregeneous multiobjective optimization. SIAM J. Optim. 29, 1017–1047 (2019)

    Article  MathSciNet  Google Scholar 

  29. Dennis Jr., J.E., Woods, D.J.: Optimization on microcomputers: the Nelder–Mead simplex algorithm. In: Wouk, A. (ed.) New Computing Environments: Microcomputers in Large-Scale Computing, pp. 116–122. SIAM, Philadelphia (1987)

    Google Scholar 

  30. Liuzzi, G., Lucidi, S., Rinaldi, F.: A derivative-free approach to constrained multiobjective nonsmooth optimization. SIAM J. Optim. 26, 2744–2774 (2016)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors would like to thank the three anonymous referees, whose comments and suggestions much improved the quality of the paper. Funding support for Ana Luísa Custódio and Rohollah Garmanjani was provided by national funds through FCT–Fundação para a Ciência e a Tecnologia I. P., under the scope of Projects PTDC/MAT-APL/28400/2017 and UIDB/00297/2020. Support for Elisa Riccietti was provided by TOTAL E&P.

Author information

Authors and Affiliations

  1. Department of Mathematics, FCT-UNL-CMA, Campus de Caparica, 2829-516, Caparica, Portugal

    Ana Luísa Custódio & Rohollah Garmanjani

  2. ISAE-SUPAERO, Université de Toulouse, 31055, Toulouse Cedex 4, France

    Youssef Diouane

  3. IRIT - ENSEEIHT, Université de Toulouse, 31071, Toulouse Cedex 7, France

    Elisa Riccietti

Authors
  1. Ana Luísa Custódio
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Youssef Diouane
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Rohollah Garmanjani
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Elisa Riccietti
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ana Luísa Custódio.

Additional information

Communicated by Xinmin Yang.

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

Custódio, A.L., Diouane, Y., Garmanjani, R. et al. Worst-Case Complexity Bounds of Directional Direct-Search Methods for Multiobjective Optimization. J Optim Theory Appl 188, 73–93 (2021). https://doi.org/10.1007/s10957-020-01781-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-020-01781-z

Keywords

Mathematics Subject Classification

Search

Navigation