Skip to main content
SpringerLink
Log in

Primal-dual subgradient method for constrained convex optimization problems

  • Short Communication
  • Published:
Optimization Letters Aims and scope Submit manuscript

Abstract

This paper considers a general convex constrained problem setting where functions are not assumed to be differentiable nor Lipschitz continuous. Our motivation is in finding a simple first-order method for solving a wide range of convex optimization problems with minimal requirements. We study the method of weighted dual averages (Nesterov in Math Programm 120(1): 221–259, 2009) in this setting and prove that it is an optimal method.

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

References

  1. Beck, A., Ben-Tal, A., Guttmann-Beck, N., Tetruashvili, L.: The CoMirror algorithm for solving nonsmooth constrained convex problems. Oper. Res. Lett. 38(6), 493–498 (2010)

    Article  MathSciNet  Google Scholar 

  2. Bertsekas, D.P.: Convex Optimization Theory. Athena Scientific, Athena (2009)

    MATH  Google Scholar 

  3. Grimmer, B.: Convergence rates for deterministic and stochastic subgradient methods without Lipschitz continuity. SIAM J. Optim. 29(2), 1350–1365 (2019)

    Article  MathSciNet  Google Scholar 

  4. Hiriart-Urruty, J.B., Lemaréchal, C.: Convex Analysis and Minimization Algorithms I: Fundamentals. Springer, Berlin (1996)

    MATH  Google Scholar 

  5. Lan, G., Zhou, Z.: Algorithms for stochastic optimization with functional or expectation constraints. Comput. Optim. Appl. 76, 461–498 (2020)

    Article  MathSciNet  Google Scholar 

  6. Nesterov, Y.: Introductory Lectures on Convex Optimization: A Basic Course. Springer, Berlin (2004)

    Book  Google Scholar 

  7. Nesterov, Y.: Primal-dual subgradient methods for convex problems. Math. Programm. 120(1), 221–259 (2009)

    Article  MathSciNet  Google Scholar 

  8. Shalev-Shwartz, S., Ben-David, S.: Understanding Machine Learning: From Theory to Algorithms. Cambridge University Press, Cambridge (2014)

    Book  Google Scholar 

  9. Xu, Y.: Primal-dual stochastic gradient method for convex programs with many functional constraints. SIAM J. Optim. 30(2), 1664–1692 (2020)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. RIKEN Center for Advanced Intelligence Project, Tokyo, Japan

    Michael R. Metel & Akiko Takeda

  2. Department of Creative Informatics, Graduate School of Information Science and Technology, The University of Tokyo, Tokyo, Japan

    Akiko Takeda

Authors
  1. Michael R. Metel
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Akiko Takeda
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Michael R. Metel.

Ethics declarations

Funding

The research of the first author is supported in part by JSPS KAKENHI Grants No. 19H04069. The research of the second author is supported in part by JSPS KAKENHI Grants No. 17H01699 and 19H04069.

Conflicts of interest

The authors declare that they have no conflict of interest.

Availability of data and material

Not applicable.

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

Metel, M.R., Takeda, A. Primal-dual subgradient method for constrained convex optimization problems. Optim Lett 15, 1491–1504 (2021). https://doi.org/10.1007/s11590-021-01728-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11590-021-01728-x

Keywords

Search

Navigation