Skip to main content
Springer Nature Link
Log in

An indefinite proximal subgradient-based algorithm for nonsmooth composite optimization

  • Published:
Journal of Global Optimization Aims and scope Submit manuscript

Abstract

We propose an indefinite proximal subgradient-based algorithm (IPSB) for solving nonsmooth composite optimization problems. IPSB is a generalization of the Nesterov’s dual algorithm, where an indefinite proximal term is added to the subproblems, which can make the subproblem easier and the algorithm efficient when an appropriate proximal operator is judiciously setting down. Under mild assumptions, we establish sublinear convergence of IPSB to a region of the optimal value. We also report some numerical results, demonstrating the efficiency of IPSB in comparing with the classical dual averaging-type algorithms.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

Data availability statements

The authors confirm that all data generated or analysed during this study are included in the paper.

References

  1. Beck, A., Teboulle, M.: A fast iterative shrinkage-thresholding algorithm for linear inverse problems. SIAM J. Imaging Sci. 2(1), 183–202 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bertsekas, D.P.: Nonlinear Programming. Taylor & Francis, Milton Park (1997)

    Google Scholar 

  3. Boyd, S., Xiao, L., Mutapcic, A.: Subgradient methods. Lecture Notes of EE392o, Stanford University, Autumn Quarter, 2004:2004–2005 (2003)

  4. Burke, J.V., Curtis, F.E., Lewis, A.S., Overton, M.L., Simões, L.E.: Gradient sampling methods for nonsmooth optimization. In: Numerical Nonsmooth Optimization, pp. 201–225. Springer (2020)

  5. Cai, X.-J., Guo, K., Jiang, F., Wang, K., Wu, Z.-M., Han, D.-R.: The developments of proximal point algorithms. J. Oper. Res. Soc. China 1–43 (2022)

  6. Combettes,P.L., Pesquet, J.C.: Proximal splitting methods in signal processing. In: Fixed-Point Algorithms for Inverse Problems in Science and Engineering, pp. 185–212. Springer (2011)

  7. Combettes, P.L., Wajs, V.R.: Signal recovery by proximal forward–backward splitting. Multiscale Model. Simul. 4(4), 1168–1200 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  8. Duchi, J., Hazan, E., Singer, Y.: Adaptive subgradient methods for online learning and stochastic optimization. J. Mach. Learn. Res. 12(7), 2021–2059 (2011)

    MathSciNet  MATH  Google Scholar 

  9. Eckstein, J., Bertsekas, D.P.: On the Douglas–Rachford splitting method and the proximal point algorithm for maximal monotone operators. Math. Program. 55(1), 293–318 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  10. Geman, S., Geman, D.: Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images. IEEE Trans. Pattern Anal. Mach. Intell. 6(6), 721–741 (1984)

    Article  MATH  Google Scholar 

  11. Grötschel, M., Lovász, L., Schrijver, A.: The ellipsoid method. In: Geometric Algorithms and Combinatorial Optimization, pp. 64–101. Springer (1993)

  12. He, B., Ma, F., Yuan, X.: Optimal proximal augmented Lagrangian method and its application to full Jacobian splitting for multi-block separable convex minimization problems. IMA J. Numer. Anal. 40(2), 1188–1216 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  13. Jiang, F., Cai, X., Han, D.: The indefinite proximal point algorithms for maximal monotone operators. Optimization 70(8), 1759–1790 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  14. Jiang, F., Wu, Z., Cai, X.: Generalized ADMM with optimal indefinite proximal term for linearly constrained convex optimization. J. Ind. Manag. Optim. 16(2), 835–856 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  15. LeCun, Y., Cortes, C., Burges, C.J.C.: The MNIST database of handwritten digits. http://yann.lecun.com/exdb/mnist/ (2017)

  16. Li, M., Sun, D., Toh, K.C.: A majorized ADMM with indefinite proximal terms for linearly constrained convex composite optimization. SIAM J. Optim. 26(2), 922–950 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  17. Mäkelä, M.: Survey of bundle methods for nonsmooth optimization. Optim. Methods Softw. 17(1), 1–29 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  18. Martinet, B.: Regularization d’inequations variationelles par approximations successives. Revue Francaise d’Informatique et de Recherche Opérationelle 4, 154–159 (1970)

    MATH  Google Scholar 

  19. Nesterov, Y.: Smooth minimization of nonsmooth functions. Math. Program. 103(1), 127–152 (2005)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  21. Nesterov, Y.: Introductory Lectures on Convex Optimization: A Basic Course, vol. 87. Springer, Berlin (2013)

    MATH  Google Scholar 

  22. Ram, S.S., Nedić, A., Veeravalli, V.V.: Incremental stochastic subgradient algorithms for convex optimization. SIAM J. Optim. 20(2), 691–717 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  23. Ram, S.S., Nedić, A., Veeravalli, V.V.: Distributed stochastic subgradient projection algorithms for convex optimization. J. Optim. Theory Appl. 147(3), 516–545 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  24. Rockafellar, R.T.: Monotone operators and the proximal point algorithm. SIAM J. Control Optim. 14(5), 877–898 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  25. Shor, N.Z.: Minimization Methods for Non-differentiable Functions. Springer Series in Computational Mathematics, Springer, Berlin (1985)

    Book  MATH  Google Scholar 

  26. Xiao, L.: Dual averaging methods for regularized stochastic learning and online optimization. J. Mach. Learn. Res. 11(10), 2543–2596 (2010)

    MathSciNet  MATH  Google Scholar 

  27. Zou, H., Hastie, T.: Regularization and variable selection via the elastic net. J. R. Stat. Soc. Ser. B 67, 301–320 (2005)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors thank the editor and the referees for the valuable comments/suggestions, which help us improve the paper greatly. The research of the second author was partially supported by NSFC with Nos. 12131004 and 12126603; and the research of the third author was partially supported by NSFC with No. 12171021 and by Beijing NSF with No. Z180005.

Author information

Authors and Affiliations

  1. LMIB, School of Mathematical Sciences, Beihang University, Beijing, 100191, China

    Rui Liu, Deren Han & Yong Xia

Authors
  1. Rui Liu
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. Deren Han
    View author publications

    You can also search for this author inPubMed Google Scholar

  3. Yong Xia
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to Deren Han.

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

Liu, R., Han, D. & Xia, Y. An indefinite proximal subgradient-based algorithm for nonsmooth composite optimization. J Glob Optim 87, 533–550 (2023). https://doi.org/10.1007/s10898-022-01173-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10898-022-01173-9

Keywords