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Dual extrapolation and its applications to solving variational inequalities and related problems

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Abstract

In this paper we suggest new dual methods for solving variational inequalities with monotone operators. We show that with an appropriate step-size strategy, our method is optimal both for Lipschitz continuous operators ( \(O({1 \over \epsilon})\) iterations), and for the operators with bounded variations ( \(O({1 \over \epsilon^2})\) iterations). Our technique can be applied for solving non-smooth convex minimization problems with known structure. In this case the worst-case complexity bound is \(O({1 \over \epsilon})\) iterations.

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References

  1. Korpelevich, G.: Extrapolation gradient methods and relation to Modified Lagrangeans. Ekonomika i Matematicheskie Metody 19, 694–703 (1983) (in Russian; English translation in Matekon)

  2. Nemirovski A. (1992): Informational-based complexity of linear operator equations. J. Complex. 8, 153–175

    Article  Google Scholar 

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

    MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  5. Nesterov Yu. (2005): Smooth minimization of non-smooth functions. Math. Program. 103(1): 127–152

    Article  MathSciNet  Google Scholar 

  6. Nesterov Yu. (2005): Excessive gap technique in non-smooth convex minimization. SIAM J. Optim. 16(1): 235–249

    Article  MathSciNet  Google Scholar 

  7. Nesterov Yu., Vial J.-Ph. (1999): Homogeneous analytic center cutting plane methods for convex problems and variational inequalities. SIAM J. Optim. 9(3): 707–728

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Center for Operations Research and Econometrics (CORE), Catholic University of Louvain (UCL), 34 voie du Roman Pays, 1348, Louvain-la-Neuve, Belgium

    Yurii Nesterov

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  1. Yurii Nesterov
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Correspondence to Yurii Nesterov.

Additional information

Dedicated to the memory of Joseph Sturm.

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 its author(s).

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Nesterov, Y. Dual extrapolation and its applications to solving variational inequalities and related problems. Math. Program. 109, 319–344 (2007). https://doi.org/10.1007/s10107-006-0034-z

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  • DOI: https://doi.org/10.1007/s10107-006-0034-z

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