Skip to main content

Primal-dual subgradient methods for convex problems

Abstract

In this paper we present a new approach for constructing subgradient schemes for different types of nonsmooth problems with convex structure. Our methods are primal-dual since they are always able to generate a feasible approximation to the optimum of an appropriately formulated dual problem. Besides other advantages, this useful feature provides the methods with a reliable stopping criterion. The proposed schemes differ from the classical approaches (divergent series methods, mirror descent methods) by presence of two control sequences. The first sequence is responsible for aggregating the support functions in the dual space, and the second one establishes a dynamically updated scale between the primal and dual spaces. This additional flexibility allows to guarantee a boundedness of the sequence of primal test points even in the case of unbounded feasible set (however, we always assume the uniform boundedness of subgradients). We present the variants of subgradient schemes for nonsmooth convex minimization, minimax problems, saddle point problems, variational inequalities, and stochastic optimization. In all situations our methods are proved to be optimal from the view point of worst-case black-box lower complexity bounds.

This is a preview of subscription content, access via your institution.

References

  1. 1.

    Auslender A. and Teboulle M. (2005). Interior projection-like methods for monotone variational inequalities. Math. Program. 104(1): 39–68

    MATH  Article  MathSciNet  Google Scholar 

  2. 2.

    Andersen S.P., de Palma A. and Thisse J.-F. (1992). Discrete choice theory of product differentiation. MIT Press, Cambridge

    Google Scholar 

  3. 3.

    Anstreicher, K., Wolsey, L.: On dual solutions in subgradient optimization. Math. Program. (in press)

  4. 4.

    Beck A. and Teboulle M. (2003). Mirror descent and nonlinear projected subgradient methods for convex optimization. Oper. Res. Lett. 31: 167–175

    MATH  Article  MathSciNet  Google Scholar 

  5. 5.

    Ben-Tal A., Margalit T. and Nemirovski A. (2001). The ordered subsets mirror descent optimization method with applications to tomography. SIOPT 12(1): 79–108

    MATH  MathSciNet  Google Scholar 

  6. 6.

    Ermoliev Yu.M. (1966). Methods for solving nonlinear extremal problems. Kibernetika 4: 1–17

    Google Scholar 

  7. 7.

    Hiriart-Urruty. J.-B., Lemarecha, C.: Convex analysis and minimization algorithms, vol. I–II. Springer, Berlin (1993)

  8. 8.

    Nemirovski A. (2004). Prox-method with rate of convergence O(1/t) for variational inequalities with Lipschitz continuous monotone operators and smooth convex-concave saddle point problems. SIOPT 15(1): 229–251

    MATH  MathSciNet  Google Scholar 

  9. 9.

    Nemirovski A. and Yudin D. (1983). Problem Complexity and Method Efficiency in Optimization. Wiley,

    Google Scholar 

  10. 10.

    Nesterov Yu. (2004). Introductory Lectures on Convex Optimization. A basic course. Kluwer, Boston

    MATH  Google Scholar 

  11. 11.

    Nesterov Yu. (2005). Smooth minimization of nonsmooth functions (CORE Discussion Paper #2003/12, CORE 2003). Math. Program. 103(1): 127–152

    MATH  Article  MathSciNet  Google Scholar 

  12. 12.

    Nesterov Yu. (2005). Excessive gap technique in nonsmooth convex minimization (CORE Discussion Paper #2003/35, CORE 2003). SIOPT 16(1): 235–249

    MATH  MathSciNet  Google Scholar 

  13. 13.

    Nesterov, Yu.: Dual extrapolation and its applications for solving variational inequalities and related problems (CORE Discussion Paper #2003/68, CORE 2003). Math. Program. doi: 10.1007/s10107-006-0034-z

  14. 14.

    Nesterov, Yu., Vial, J.-Ph.: Confidence level solutions for stochastic programming. CORE Discussion Paper #2000/13, CORE 2000

  15. 15.

    Ortega J.M. and Reinboldt W.C. (1970). Iterative Solution of Nonlinear Equations in Several Variables. Academic Press, New York

    MATH  Google Scholar 

  16. 16.

    Polyak B.T. (1967). A general method of solving extremum problems. Soviet Mat. Dokl. 8: 593–597

    MATH  Google Scholar 

  17. 17.

    Shor N.Z. (1985). Minimization Methods for Nondifferentiable Functions. Springer, Berlin

    Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Yurii Nesterov.

Additional information

Dedicated to B. T. Polyak on the occasion of his 70th birthday

The research results presented in this paper have been supported by a grant “Action de recherche concertè ARC 04/09-315” from the “Direction de la recherche scientifique - Communautè française de Belgique”. The scientific responsibility rests with the author.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Nesterov, Y. Primal-dual subgradient methods for convex problems. Math. Program. 120, 221–259 (2009). https://doi.org/10.1007/s10107-007-0149-x

Download citation

Keywords

  • Convex optimization
  • Subgradient methods
  • Non-smooth optimization
  • Minimax problems
  • Saddle points
  • Variational inequalities
  • Stochastic optimization
  • Black-box methods
  • Lower complexity bounds

Mathematics Subject Classification (2000)

  • 90C25
  • 90C47
  • 68Q25