nach oben

Journal of Scientific Computing

Erschienen in:

01.10.2020

A New Method for Solving Variational Inequalities and Fixed Points Problems of Demi-Contractive Mappings in Hilbert Spaces

verfasst von: Xue Chen, Zhong-bao Wang, Zhang-you Chen

Erschienen in: Journal of Scientific Computing | Ausgabe 1/2020

Einloggen

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config

KI-gestützte Suche

Aus

Abstract

In this paper, an efficient algorithm for solving variational inequalities and fixed points problems of demi-contractive mappings is proposed in Hilbert spaces. The algorithm uses variable stepsizes which are updated at each iteration by a cheap computation without any linesearch procedure. Under the assumption that the mapping is pseudomonotone and without prior knowledge of the Lipschitz constant of the underlying operator, the sequence generated by the algorithm is strongly convergent to a common element of the set of fixed points of a demi-contractive mapping and the solution set of variational inequalities. Some experiments are performed to show the numerical behavior of the proposed algorithm and also to compare its performance with those of others.

Vorheriger Artikel Compact Schemes for Multiscale Flows with Cell-Centered Finite Difference Method

Nächster Artikel A Parareal Finite Volume Method for Variable-Order Time-Fractional Diffusion Equations

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

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

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

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

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 340 Zeitschriften

aus folgenden Fachgebieten:

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

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 390 Zeitschriften

aus folgenden Fachgebieten:

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

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Facchinei, F., Pang, J.S.: Finite-dimensional Variational Inequalities and Complementarity Problems, vol. I. Springer, New York (2003)MATH

Iusem, A.N., Svaiter, B.F.: A variant of Korpelevich’s method for variational inequalities with a new search strategy. Optimization 42(4), 309–321 (1997)MathSciNetCrossRef

Solodov, M.V., Svaiter, B.F.: A new projection method for variational inequality problems. SIAM J. Control Optim. 37(3), 765–776 (1999)MathSciNetCrossRef

Ye, M.L., He, Y.R.: A double projection method for solving variational inequalities without mononicity. Comput. Optim. Appl. 60(1), 141–150 (2015)MathSciNetCrossRef

Fang, C.J., He, Y.R.: An extragradient method for generalized variational inequality. Pac. J. Optim. 9, 47–59 (2013)MathSciNetMATH

Censor, Y., Gibali, A., Reich, S.: The subgradient extragradient method for solving variational inequalities in Hilbert space. J. Optim. Theory Appl. 148, 318–335 (2011)MathSciNetCrossRef

Sibony, M.: Methodes iteratives pour les equation set enequalitions aux derivees partielles nonlinearesde type monotone. Calcolo 7, 65–183 (1970)MathSciNetCrossRef

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

Antipin, A.S.: On a method for convex programs using a symmetrical modification of the Lagrange function. Ekonomika i Matematicheskie Metody 12(6), 1164–1173 (1976)

10.

Censor, Y., Gibali, A., Reich, S.: The subgradient extragradient method for solving variational inequalities in Hilbert spaces. J. Optim. Theory Appl. 148, 318–335 (2011)MathSciNetCrossRef

11.

Thong, D.V., Vinh, N.T., Cho, Y.J.: Accelerated subgradient extragradient methods for variational inequality problems. J. Sci. Comput. 80, 1436–1462 (2019)MathSciNetCrossRef

12.

Malitsky, Y.V.: Projected reflected gradient methods for variational inequalities. SIAM J. Optim. 25, 502–520 (2015)MathSciNetCrossRef

13.

Hieu, D.V., Anh, P.K., Muu, L.D.: Modified extragradient-like algorithms with new stepsizes for variational inequalities. Comput. Optim. Appl. 73, 913–932 (2019)MathSciNetCrossRef

14.

Hieu, D.V., Cho, Y.J., Xiao, Y.B.: Golden ratio algorithms with new stepsize rules for variational inequalities. Math. Methods Appl. 42, 6067–6082 (2019)MathSciNetCrossRef

15.

Hieu, D.V., Cho, Y.J., Xiao, Y.B.: Modified extragradient algorithms for solving equilibrium problems. Optimization 67, 2003–2029 (2018)MathSciNetCrossRef

16.

Wang, Z.B., Chen, Z.Y., Xiao, X.B., Zhang, C.: A new projection-type method for solving multi-valued mixed variational inequalities without monotonicity. Appl. Anal. 99(9), 1453–1466 (2020)MathSciNetCrossRef

17.

Takahashi, W.: Convex Analysis and Approximation of Fixed Points. Yokohama Publishers, Yokohama (2000)MATH

18.

Yamada, I.: The hybrid steepest descent method for the variational inequality over the intersection of fixed point sets of nonexpansive mappings. in Inherently Parallel Algorithms in Feasibility and Optimization and Their Applications, D. Butnariu, Y. Censor, and S. Reich, eds., Elsevier, Amsterdam, pp. 473–504 (2001)

19.

Zeidler, E.: Nonlinear Functional Analysis and Its Applications. Springer, New York (1985)CrossRef

20.

Mainge, P.E.: A hybrid extragradient-viscosity method for monotone operators and fixed point problems. SIAM J. Control Optim. 47(3), 1499–1515 (2008)MathSciNetCrossRef

21.

Ceng, L.C., Hadjisavvas, N., Wong, N.C.: Strong convergence theorem by a hybrid extragradientlike approximation method for variational inequalities and fixed point problems. J. Glob. Optim. 46, 635–646 (2010)CrossRef

22.

Thong, D.V., Hieu, D.V.: Inertial subgradient extragradient algorithms with line-search process for solving variational inequality problems and fixed point problems. Numer. Algor. 80(4), 1283–1307 (2019)MathSciNetCrossRef

23.

Gibali, A., Shehu, Y.: An efficient iterative method for finding common fixed point and variational inequalities in Hilbert spaces. Optimization 68(1), 13–32 (2019)MathSciNetCrossRef

24.

Iiduka, H., Yamada, I.: A use of conjugate gradient direction for the convex optimization problem over the fixed point set of a nonexpansive mapping. SIAM J. Optim. 19, 1881–1893 (2008)MathSciNetCrossRef

25.

Zhang, L., Fang, C., Chen, S.: An inertial subgradient-type method for solving single-valued variational inequalities and fixed point problems. Numer. Algor. 79, 941–956 (2018)MathSciNetCrossRef

26.

Vuong, P.T., Shehu, Y.: Convergence of an extragradient-type method for variational inequality with applications to optimal control problems. Numer. Algor. 81(1), 269–291 (2018)MathSciNetCrossRef

27.

Malitsky, Y.: Golden ratio algorithms for variational inequalities. Math. Prog. https://doi.org/10.1007/s10107-019-01416-w

28.

Goebel, K., Reich, S.: Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings. Marcel Dekker, New York (1984)MATH

29.

Zarantonello, E.H.: Projections on Convex Sets in Hilbert space and Spectral theory. In: Zarantonello, E.H. (ed.) Contributions to Nonlinear Functional Analysis. Academic Press, New York (1971)MATH

30.

Xu, H.K.: Iterative algorithms for nonlinear operators. J. Lond. Math. Soc. 66(1), 240–256 (2002)MathSciNetCrossRef

31.

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

32.

Mashreghi, J., Nasri, M.: Forcing strong convergence of Korpelevich\(!^{-}\) method in Banach spaces with its applications in game theory. Nonlinear Anal. 72, 2086–2099 (2010)MathSciNetCrossRef

33.

Dong, Q.L., Lu, Y.Y., Yang, J.: The extragradient algorithm with inertial effects for solving the variational inequality. Optimization 65(12), 2217–2226 (2016)MathSciNetCrossRef

34.

Solodov, M.V., Svaiter, B.F.: A new projection method for variational inequality problems. SIAM J. Control Optim. 37, 765–776 (1999)MathSciNetCrossRef

35.

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

36.

Hu, X., Wang, J.: Solving pseudo-monotone variational inequalities and pseudo-convex optimization problems using the projection neural network. IEEE Trans. Neural Netw. 17, 1487–1499 (2006)CrossRef

Titel: A New Method for Solving Variational Inequalities and Fixed Points Problems of Demi-Contractive Mappings in Hilbert Spaces
verfasst von: Xue Chen
Zhong-bao Wang
Zhang-you Chen
Publikationsdatum: 01.10.2020
Verlag: Springer US
Erschienen in: Journal of Scientific Computing / Ausgabe 1/2020
Print ISSN: 0885-7474
Elektronische ISSN: 1573-7691
DOI: https://doi.org/10.1007/s10915-020-01327-5

Springer Professional

Abstract

Bitte loggen Sie sich ein, um Zugang zu Ihrer Lizenz zu erhalten.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Springer Professional "Wirtschaft"

Springer Professional "Technik"

Weitere Artikel der Ausgabe 1/2020

A Divergence-Conforming DG-Mixed Finite Element Method for the Stationary Boussinesq Problem

A Parareal Finite Volume Method for Variable-Order Time-Fractional Diffusion Equations

Admissible Concentration Factors for Edge Detection from Non-uniform Fourier Data

A Novel Arbitrary Lagrangian–Eulerian Finite Element Method for a Mixed Parabolic Problem in a Moving Domain

Compact Schemes for Multiscale Flows with Cell-Centered Finite Difference Method

A New Projection-Based Stabilized Virtual Element Method for the Stokes Problem

Premium Partner