Skip to main content
SpringerLink
Log in

Adaptive Conditional Gradient Method

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

Abstract

We present a novel fully adaptive conditional gradient method with the step length regulation for solving pseudo-convex constrained optimization problems. We propose some deterministic rules of the step length regulation in a normalized direction. These rules guarantee to find the step length by utilizing the finite procedures and provide the strict relaxation of the objective function at each iteration. We prove that the sequence of the function values for the iterates generated by the algorithm converges globally to the objective function optimal value with sublinear rate.

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.

Similar content being viewed by others

References

  1. Dunn, J.C., Hafghbarger, S.: Conditional gradient algorithms with open loop step size rules. J. Math. Anal. Appl. 62, 432–444 (1978)

    Article  MathSciNet  Google Scholar 

  2. Lacoste-Julien, S., Jaggi, M., Schmidt, M., Pletscher, P.: Block-coordinate Frank–Wolfe optimization for structural SVMs. In: Proceedings of the 30th International Conference on Machine Learning, PMLR, vol. 28(1), pp. 53–61 (2013)

  3. Braun, G., Pokutta, S., Zink, D.: Lazifying conditional gradient algorithms. arXiv preprint arXiv:1610.051220v4 (2018)

  4. Lacoste-Julien, S.: Convergence rate of Frank–Wolfe for non-convex objectives. arXiv preprint arXiv:1607.00345 (2016)

  5. Nesterov, Y.: Complexity bounds for primal-dual methods minimizing the model of objective function. Math. Program. 171(1–2), 311–330 (2018)

    Article  MathSciNet  Google Scholar 

  6. Bertsekas, D.P.: Nonlinear Programming, 2nd edn. Athena Scientific, Belmont (1999)

    MATH  Google Scholar 

  7. Yu, Y., Zhang, X., Schuurmans, D.: Generalized conditional gradient for sparse estimation. arXiv preprint arXiv:1410.4828v1 (2014)

  8. Gabidullina, Z.R.: Relaxation methods with step regulation for solving constrained optimization problems. J. Math. Sci. 73(5), 538–543 (1995)

    Article  MathSciNet  Google Scholar 

  9. Mitchell, V.F., Dem’yanov, V.F., Malozemov, V.N.: Finding the point of polyhedron closest to origin. SIAM J. Control Optim. 12, 19–26 (1974)

    Article  MathSciNet  Google Scholar 

  10. Gabidullina, Z.R.: The problem of projecting the origin of euclidean space onto the convex polyhedron. Lobachevskii J. Math. 39(1), 35–45 (2018)

    Article  MathSciNet  Google Scholar 

  11. Gabidullina, Z.R.: Solving of a projection problem for convex polyhedra given by a system of linear constraints. In: Constructive Nonsmooth Analysis and Related Topics (Dedicated to the Memory of V.F. Demyanov), CNSA Proceedings—IEEE, Art. 7973958 (2017). http://ieeexplore.ieee.org/abstract/document/7973958/

  12. Gabidullina, Z.R.: The Minkowski difference for convex polyhedra and some its applications. arXiv preprint arXiv:1903.03590 (2019)

  13. Mangasarian, O.L.: Pseudo-convex functions. J. Soc. Ind. Appl. Math. Ser. A Control 3, 281–290 (1965)

    Article  MathSciNet  Google Scholar 

  14. Gabidullina, Z.R.: Convergence of the constrained gradient method for a class of nonconvex functions. J. Sov. Math. 50(5), 1803–1809 (1990)

    Article  MathSciNet  Google Scholar 

  15. Gabidullina, Z.R.: Adaptive methods with step length regulation for solving pseudo-convex programming problems. Dissertation for the Degree of Candidate of Science in Physics and Mathematics, Kazan (1994)

  16. Vasil’ev, F.P.: Numerical Methods for Solving Extremum Problems. Nauka, Moscow (1980)

    Google Scholar 

  17. Reklaitis, G.V., Ravindran, A., Ragsdell, K.M.: Engineering Optimization: Methods and Applications, 2nd edn. Wiley, Hoboken (2006)

    Google Scholar 

  18. Beltukov, I.B., Shurygina, M.N.: A study of one adaptive method for mathematical programming. In: Optimization Methods and Applications, pp. 5–13. Irkutsk (1988) (in Russian)

Download references

Acknowledgements

The author thanks the anonymous referees and the editor for their helpful comments and remarks on a previous version of the paper.

Author information

Authors and Affiliations

  1. Department of Data Analysis and Operations Research, Kazan Federal University, 18, Kremljovskaja Street, Kazan, Russia, 420008

    Z. R. Gabidullina

Authors
  1. Z. R. Gabidullina
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Z. R. Gabidullina.

Additional information

Alexanre Cabot.

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

Gabidullina, Z.R. Adaptive Conditional Gradient Method. J Optim Theory Appl 183, 1077–1098 (2019). https://doi.org/10.1007/s10957-019-01585-w

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-019-01585-w

Keywords

Mathematics Subject Classification

Search

Navigation