Library Subscription: Guest
Begell Digital Portal Begell Digital Library eBooks Journals References & Proceedings Research Collections
Journal of Automation and Information Sciences
SJR: 0.238 SNIP: 0.464 CiteScore™: 0.27

ISSN Print: 1064-2315
ISSN Online: 2163-9337

Volumes:
Volume 51, 2019 Volume 50, 2018 Volume 49, 2017 Volume 48, 2016 Volume 47, 2015 Volume 46, 2014 Volume 45, 2013 Volume 44, 2012 Volume 43, 2011 Volume 42, 2010 Volume 41, 2009 Volume 40, 2008 Volume 39, 2007 Volume 38, 2006 Volume 37, 2005 Volume 36, 2004 Volume 35, 2003 Volume 34, 2002 Volume 33, 2001 Volume 32, 2000 Volume 31, 1999 Volume 30, 1998 Volume 29, 1997 Volume 28, 1996

Journal of Automation and Information Sciences

DOI: 10.1615/JAutomatInfScien.v50.i8.30
pages 26-37

Modified Extragradient Method with Bregman Divergence for Variational Inequalities

Vladimir V. Semenov
Kiev National Taras Shevchenko University, Kiev

ABSTRACT

A new method of extragradient type for the approximate solution of variational inequalities with pseudomonotone and Lipschitz-continuous operators acting in a finite-dimensional linear normed space is proposed. This method is a modification of the subgradient extragradient algorithm using Bregman divergence instead of Euclidean distance. Like other schemes using Bregman divergence the proposed method can sometimes effectively take into account a structure of a feasible set of the problem. The theorem on the method convergence is proved and in the case of a monotone operator nonasymptotic estimates of the method effectiveness are obtained

REFERENCES

  1. Konnov I.V., Combined relaxation methods for variational inequalities, Springer-Verlag, Berlin-Heidelberg-New York, 2001.

  2. Korpelevich G.M., Extragradient method for finding saddle points and other problems, Ekonomika i matematicheskie metody, 1976, 12, No. 4, 747–756.

  3. Khobotov E.N., On modification of extragradient method for solving variational inequalities and some optimization problems, Zhurnal vychislitelnoy matematiki i matematicheskoy fiziki, 1987, No. 10, 1462–1473.

  4. Tseng P., A modified forward-backward splitting method for maximal monotone mappings, SIAM Journal on Control and Optimization, 2000, 38, 431–446.

  5. Semenov V.V., Hybrid splitting methods for systems of operator inclusions with monotone operators, Kibernetika i sistemnyi analiz, 2014, No. 5, 104–112.

  6. Semenov V.V., Strongly converging splitting method for systems of operator inclusions with monotone operators, Mezhdunarodnyi nauchno-tekhnicheskiy zhurnal “Problemy upravleniya i informatiki”, 2014, No. 3, 22–32.

  7. Lyashko S.I., Semenov V.V., A new two-step proximal algorithm of solving the problem of equilibrium programming, In: Goldengorin B. (ed.) Optimization and its Applications in Control and Data Sciences, Springer Optimization and its Applications, Cham, Springer, 2016, 115, 315–325.

  8. Censor Y., Gibali A., Reich S., The subgradient extragradient method for solving variational inequalities in Hilbert space, Journal of Optimization Theory and Applications, 2011, 148, 318–335.

  9. Lyashko S.I., Semenov V.V., Voytova T.A., Economic modification of Korpelevich method for monotone equilibrium problems, Kibernetika i sistemnyi analiz, 2011, No. 4, 146–154.

  10. Verlan D.A., Semenov V.V., Chaback L.M., Strongly converging modified extragradient method for variational inequalities with non-Lipschitz operators, Mezhdunarodnyi nauchno-tekhnicheskiy zhurnal “Problemy upravleniya i informatiki”, 2015, No. 4, 37–50.

  11. Denisov S.V., Semenov V.V., Chaback L.M., Convergence of modified extragradient method for variational inequalities with non-Lipschitz operators, Kibernetika i sistemnyi analiz, 2015, No. 5, 102–110.

  12. Bregman L.M., Relaxation method for finding a common point of convex sets and its application to solving problems of convex programming, Zhurnal vychislit. matematiki i mat. fiziki, No. 3, 620–631.

  13. Beck A., Teboulle M., Mirror descent and nonlinear projected subgradient methods for convex optimization, Operations Research Letters, 2003, 31, No. 3, 167–175.

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

  15. Auslender A., Teboulle M., Interior projection-like methods for monotone variational inequalities, Mathematical Programming, 2005, 104, No. 1, 39–68.

  16. Nesterov Yu., Dual extrapolation and its applications to solving variational inequalities and related problems, Ibid., 2007, 109, No. 2–3, 319–344.

  17. Semenov V.V., Variant of mirror descent method for variational inequalities, Kibernetika i sistemnyi analiz, 2017, No. 2, 83–93.

  18. Semenov V.V., A variant of mirror descent method for solving variational inequalities. In: Polyakova, L.N. (ed.) Constructive Nonsmooth Analysis and Related Topics (dedicated to the memory of V.F. Demyanov) (CNSA), 2017. doi: 10.1109/CNSA.2017.7974011.

  19. Viyugin V.V., Mathematical foundations of machine learning and forecasting [in Russian], MTSNMO, Moscow, 2018.

  20. Lorenz D.A., Schöpfer F., Wenger S., The linearized Bregman method via split feasibility problems: Analysis and Generalizations, SIAM Journal on Imaging Sciences, 2014, 7, No. 2, 1237–1262.


Articles with similar content:

Convergence of Extragradient Algorithm with Monotone Step Size Strategy for Variational Inequalities and Operator Equations
Journal of Automation and Information Sciences, Vol.51, 2019, issue 6
Dmitriy A. Nomirovskii, Sergey V. Denisov, Bogdan V. Rublyov , Vladimir V. Semenov
A Strongly Convergent Modified Extragradient Method for Variational Inequalities with Non-Lipschitz Operators
Journal of Automation and Information Sciences, Vol.47, 2015, issue 7
Dmitriy A. Verlan , Lyubov M. Chabak , Vladimir V. Semenov
Analysis of a Convergence for Iterative Procedures on the Basis of Methods of Practical Stability
Journal of Automation and Information Sciences, Vol.31, 1999, issue 7-9
Alexander N. Bashnyakov, Fedor G. Garashchenko
Lyapunov Functions for Systems Satisfying Barbashin−Krasovskii Theorem Conditions
Journal of Automation and Information Sciences, Vol.40, 2008, issue 11
Alexey S. Suykov, Alexander M. Kovalev
Mathematical Model of Recognition Systems with Small Databases
Journal of Automation and Information Sciences, Vol.39, 2007, issue 10
Vitaliy A. Tayanov, Boris E. Kapustiy, Bogdan P. Rusyn