Top

previous article An arc-search predictor-corrector infeasible-in...

next article Convergence of a hybrid viscosity approximation...

Hint

Swipe to navigate through the articles of this issue

08-06-2019 | Original Paper | Issue 4/2020

Strong convergence of an extragradient-like algorithm involving pseudo-monotone mappings

Journal:: Numerical Algorithms > Issue 4/2020

Authors:: Liya Liu, Xiaolong Qin

» Get access to the full-text

Important notes

Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Abstract

The purpose of this paper is to introduce a new extragradient-like algorithm for solving a variational inequality problem with a pseudo-monotone and Lipschitz continuous mapping in a Hilbert space. The iterative algorithm combines inertial ideas and hybrid extragradient ideas with the Armijo-like step size rule. Strong convergence of the algorithm is obtained and numerical experiments are provided.

Please log in to get access to this content

To get access to this content you need the following product:

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

über 58.000 Bücher
über 300 Zeitschriften

aus folgenden Fachgebieten:

Bauwesen + Immobilien
Business IT + Informatik
Finance + Banking
Management + Führung
Marketing + Vertrieb
Versicherung + Risiko

Testen Sie jetzt 30 Tage kostenlos.

inform now

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

über 50.000 Bücher
über 380 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Umwelt
Maschinenbau + Werkstoffe

Testen Sie jetzt 30 Tage kostenlos.

inform now

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 69.000 Bücher
über 500 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Umwelt
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Testen Sie jetzt 30 Tage kostenlos.

inform now

Takahashi, W., Wen, C.F., Yao, J.C.: The shrinking projection method for a finite family of demimetric mappings with variational inequality problems in a Hilbert space. Fixed Point Theory 19, 407–419 (2018) MathSciNetCrossRef

Zhao, X., et al.: Linear regularity and linear convergence of projection-based methods for solving convex feasibility problems. Appl. Math. Optim. 78, 613–641 (2018) MathSciNetCrossRef

Zhao, J., Zong, H.: Iterative algorithms for solving the split feasibility problem in Hilbert spaces. J. Fixed Point Theory Appl. 20, 11 (2018) MathSciNetCrossRef

Chang, S.S., Wen, C.F., Yao, J.C.: Common zero point for a finite family of inclusion problems of accretive mappings in Banach spaces. Optimization 67, 1183–1196 (2018) MathSciNetCrossRef

Ansari, Q.H., Babu, F., Yao, J.C.: Regularization of proximal point algorithms in Hadamard manifolds. J. Fixed Point Theory Appl. 21, 25 (2019) MathSciNetCrossRef

Korpelevich, G.M.: The extragradient method for finding saddle points and other problems. Ekon. Mat. Metody 12, 747–756 (1976) MathSciNetMATH

Gibali, A.: A new Bregman projection method for solving variational inequalities in Hilbert spaces. Pure Appl. Funct. Anal. 3, 403–415 (2018) MathSciNet

Dong, Q., Jiang, D., Gibali, A.: A modified subgradient extragradient method for solving the variational inequality problem. Numer. Algor. 79, 927–940 (2018) MathSciNetCrossRef

Censor, Y., Gibali, A., Reich, S.: Strong convergence of subgradient extragradient methods for the variational inequality problem in Hilbert space. Optim. Methods Softw. 26, 827–845 (2011) MathSciNetCrossRef

10.

Malitsky, Y.V., Semenov, V.V.: A hybrid method without extrapolation step for solving variational inequality problems. J. Global Optim. 61, 193–202 (2015) MathSciNetCrossRef

11.

Cho, S.Y.: Strong convergence analysis of a hybrid algorithm for nonlinear operators in a Banach space. J. Appl. Appl. Comput. 8, 19–31 (2018) MathSciNet

12.

Shang, M.: A descent-like method for fixed points and split conclusion problems. J. Appl. Numer. Optim. 1, 91–101 (2019)

13.

Tseng, P.: A modified forward backward splitting method for maximal monotone mappings. SIAM J. Control Optim. 38, 431–446 (2000) MathSciNetCrossRef

14.

Dautray, R., Lions, J.L.: Mathematical Analysis and Numerical Methods for Science and Technology. Springer, New York (1988) CrossRef

15.

Khan, M.A., Yannelis, N.C.: Equilibrium Theory in Infinite Dimensional Spaces. Springer, New York (1991) CrossRef

16.

Combettes, P.L.: The Convex Feasibility Problem in Image Recovery. In: Hawkes, P (ed.) Advanced in Imaging and Electron Physcis, vol. 95, pp 155–270. Academic Press, New York (1996)

17.

Fattorini, H.O.L.: Infinite-Dimensional Optimization and Control Theory. Cambridge University Press, Cambridge (1999) CrossRef

18.

Nadezhkina, N., Takahashi, W.: Strong convergence theorem by a hybrid method for nonexpansive mappings and Lipschitz-continuous monotone mappings. SIAM J. Optim. 16, 1230–1241 (2006) MathSciNetCrossRef

19.

Polyak, B.T.: Some methods of speeding up the convergence of iteration methods. Zh. Vychisl. Mat. Mat. Fiz. 4, 1–17 (1964)

20.

Alvarez, F., Attouch, H.: An inertial proximal method for maximal monotone operators via discretization of a nonlinear oscillator with damping. Set Valued Anal. 9, 3–11 (2001) MathSciNetCrossRef

21.

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

22.

Maingé, P. E.: Regularized and inertial algorithms for common fixed points of nonlinear operators. J. Math. Anal. Appl. 34, 876–887 (2008) MathSciNetCrossRef

23.

Attouch, H., Peypouquet, J., Redont, P.: A dynamical approach to an inertial forward-backward algorithm for convex minimization. SIAM J. Optim. 24, 232–256 (2014) MathSciNetCrossRef

24.

Thong, D.V., Hieu, D.V.: Inertial extragradient algorithms for strongly pseudomonotone variational inequalities. J. Comput. Appl. Math. 341, 80–98 (2018) MathSciNetCrossRef

25.

Zhang, L., Zhao, H., Lv, Y.: A modified inertial projection and contraction algorithms for quasi-variational inequalities. Appl. Set-Val. Anal. Optim. 1, 63–76 (2019)

26.

Thong, D.V., Hieu, D.V.: Modified subgradient extragradient method for variational inequality problems. Numer. Algor. 79, 597–610 (2018) MathSciNetCrossRef

27.

Gibali, A.: A new non-Lipschitzian projection method for solving variational inequalities in Euclidean spaces. J. Nonlinear Anal. Optim. 6, 41–51 (2015) MathSciNetMATH

28.

Cioranescu, I.: Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems. Kluwer, Dordrecht (1990) CrossRef

29.

Cottle, R.W., Yao, J.C.: Pseudo-monotone complementarity problems in Hilbert space. J. Optim. Theory Appl. 75, 281–295 (1992) MathSciNetCrossRef

30.

Bauschke, H.H., Combettes, P.L.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces. Springer, New York (2011) CrossRef

Title: Strong convergence of an extragradient-like algorithm involving pseudo-monotone mappings
Authors:: Liya Liu
Xiaolong Qin
Publication date: 08-06-2019
DOI: https://doi.org/10.1007/s11075-019-00737-3
Publisher: Springer US
Journal: Numerical Algorithms
Issue 4/2020
Print ISSN: 1017-1398
Electronic ISSN: 1572-9265

Premium Partner

Image Credits

Springer Professional

Hint

Swipe to navigate through the articles of this issue

Publisher’s note

Abstract

Please log in to get access to this content

To get access to this content you need the following product:

Springer Professional "Wirtschaft"

Springer Professional "Technik"

Springer Professional "Wirtschaft+Technik"

Other articles of this Issue 4/2020

Symbolic-numeric integration of rational functions

Computing the roots of sparse high–degree polynomials that arise from the study of random simplicial complexes

Computational science in the eighteenth century. Test cases for the methods of Newton, Raphson, and Halley: 1685 to 1745

An arc-search predictor-corrector infeasible-interior-point algorithm for P∗(κ)-SCLCPs

A new algorithm that generates the image of the attractor of a generalized iterated function system

A decoupling two-grid method for the time-dependent Poisson-Nernst-Planck equations

Premium Partner