Skip to main content
SpringerLink
Log in

A primal–dual prediction–correction algorithm for saddle point optimization

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

Abstract

In this paper, we introduce a new primal–dual prediction–correction algorithm for solving a saddle point optimization problem, which serves as a bridge between the algorithms proposed in Cai et al. (J Glob Optim 57:1419–1428, 2013) and He and Yuan (SIAM J Imaging Sci 5:119–149, 2012). An interesting byproduct of the proposed method is that we obtain an easily implementable projection-based primal–dual algorithm, when the primal and dual variables belong to simple convex sets. Moreover, we establish the worst-case \({\mathcal {O}}(1/t)\) convergence rate result in an ergodic sense, where t represents the number of iterations.

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

Similar content being viewed by others

References

  1. Arrow, K., Hurwicz, L., Uzawa, H.: With contributions by H.B. Chenery, S.M. Johnson, S. Karlin, T. Marschak, and R.M. Solow. Studies in Linear and Non-Linear Programming. Stanford Mathematical Studies in the Social Sciences, vol. II. Stanford Unversity Press, Stanford (1958)

  2. Cai, X., Han, D., Xu, L.: An improved first-order primal-dual algorithm with a new correction step. J. Glob. Optim. 57, 1419–1428 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  3. Chambolle, A., Pock, T.: On the ergodic convergence rates of a first order primal dual algorithm. Math. Program. Ser. A. doi:10.1007/s10107-015-0957-3

  4. Chambolle, A., Pock, T.: A first-order primal-dual algorithm for convex problems with applications to imaging. J. Math. Imaging Vis. 40, 120–145 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  5. Chen, Y., Lan, G., Ouyang, Y.: Optimal primal dual methods for a class of saddle point problems. SIAM J. Optim. 24, 1779–1814 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  6. Esser, E., Zhang, X., Chan, T.: A general framework for a class of first-order primal-dual algorithms for convex optimization in imaging sciences. SIAM J. Imaging Sci. 3, 1015–1046 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  7. Goldstein, T., Li, M., Yuan, X., Esser, E., Baraniuk, R.: Adaptive primal-dual hybrid gradient methods for saddle point problems (2015). arXiv:1305.0546v2

  8. Gu, G., He, B., Yuan, X.: Customized proximal point algorithms for linearly constrained convex minimization and saddle-point problems: a uniform approach. Comput. Optim. Appl. 59, 135–161 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  9. Han, D., Xu, W., Yang, H.: An operator splitting method for variational inequalities with partially unknown mappings. Numer. Math. 111, 207–237 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  10. He, B., Yuan, X.: Convergence analysis of primal-dual algorithms for a saddle-point problem: from contraction perspective. SIAM J. Imaging Sci. 5, 119–149 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  11. He, B.S., You, Y., Yuan, X.M.: On the convergence of primal dual hybrid gradient algorithm. SIAM J. Imaging Sci. 7, 2526–2537 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  12. Komodakis, N., Pesquet, J.C.: Playing with duality: an overview of recent primal dual approaches for solving large scale optimization problems. IEEE Signal Process Mag. 32(6), 31–54 (2015)

    Article  Google Scholar 

  13. Nemirovski, A.: Prox-method with rate of convergence \({O}(1/t)\) for variational inequalities with Lipschitz continuous monotone operator and smooth convex-concave saddle point problems. SIAM J. Optim. 15, 229–251 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  14. Tian, W., Yuan, X.: Linearized primal-dual methods for linear inverse problems with total variation regularization and finite element discretization, Working Paper (2015). http://www.math.hkbu.edu.hk/~xmyuan/Paper/LPD-TV-June19.pdf

  15. Zhu, M., Chan, T.: An Efficient Primal-Dual Hybrid Gradient Algorithm for Total Variation Image Restoration. CAM Reports 08-34, UCLA (2008)

Download references

Acknowledgments

H.J. He was supported in part by National Natural Science Foundation of China (Grant Nos. 11301123, 71471051, and 11571087) and the Zhejiang Provincial Natural Science Foundation Grant No. LZ14A010003. J. Desai was supported in part by the Ministry of Education (Singapore) AcRF Tier 1 Grant No. M4011083.

Author information

Authors and Affiliations

  1. Department of Mathematics, School of Science, Hangzhou Dianzi University, Hangzhou, 310018, China

    Hongjin He

  2. School of Mechanical and Aerospace Engineering, Nanyang Technological University, 50 Nanyang Avenue, Singapore, 639798, Singapore

    Jitamitra Desai & Kai Wang

Authors
  1. Hongjin He
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jitamitra Desai
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Kai Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jitamitra Desai.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

He, H., Desai, J. & Wang, K. A primal–dual prediction–correction algorithm for saddle point optimization. J Glob Optim 66, 573–583 (2016). https://doi.org/10.1007/s10898-016-0437-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10898-016-0437-1

Keywords

Search

Navigation