Weak convergence of a modified subgradient extragradient algorithm for monotone variational inequalities in Banach spaces

Volume 10, Issue 10, pp 5483--5494 http://dx.doi.org/10.22436/jnsa.010.10.30

Publication Date: October 28, 2017 Submission Date: March 26, 2017

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Authors

Ying Liu - College of Mathematics and Information Science, Hebei University, Baoding, Hebei, 071002, China. Hang Kong - College of Mathematics and Information Science, Hebei University, Baoding, Hebei, 071002, China.

Abstract

Applying the generalized projection operator, we introduce a modified subgradient extragradient algorithm in Banach spaces for a variational inequality involving a monotone Lipschitz continuous mapping which is more general than an inverse-strongly-monotone mapping. Weak convergence of the iterative algorithm is also proved. An advantage of the algorithm is the computation of only one value of the inequality mapping and one projection onto the admissible set per one iteration.

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Keywords

• Subgradient extragradient method
• generalized projection operator
• monotone mapping
• variational inequality
• Lipschitz continuous
• weakly sequentially continuous

MSC

•  47H09
•  47H05
•  47H06
•  47J25
•  47J05

References

• [1] Y. I. Al’ber, S. Reich , An iterative method for solving a class of nonlinear operator equations in Banach spaces , Panamer. Math. J., 4 (1994), 39–54.
  • MathSciNet
  • Google Scholar


• [2] K. Ball, E. A. Carlen, E. H. Lieb, Sharp uniform convexity and smoothness inequalities for trace norms, Invent. Math., 115 (1994), 463–482.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [3] N. Buong , Strong convergence theorem of an iterative method for variational inequalities and fixed point problems in Hilbert spaces , Appl. Math. Comput., 217 (2010), 322–329.
  • View Article
  • MathSciNet
  • Google Scholar


• [4] L.-C. Ceng, N. Hadjisavvas, N.-C. Wong , Strong convergence theorem by a hybrid extragradient-like approximation method for variational inequalities and fixed point problems , J. Global Optim., 46 (2010), 635–646.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [5] Y. Censor, A. Gibali, S. Reich, Strong convergence of subgradient extragradient methods for the variational inequality problem in Hilbert space, Optim. Methods Softw., 26 (2011), 827–845.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [6] Y. Censor, A. Gibali, S. Reich, The subgradient extragradient method for solving variational inequalities in Hilbert space , J. Optim. Theory Appl., 148 (2011), 318–335.
  • View Article
  • MathSciNet
  • Google Scholar


• [7] J.-M. Chen, L.-J. Zhang, T.-G. Fan, Viscosity approximation methods for nonexpansive mappings and monotone mappings, J. Math. Anal. Appl., 334 (2007), 1450–1461.
  • View Article
  • Google Scholar


• [8] C.-J. Fang, Y.Wang, S.-K. Yang , Two algorithms for solving single-valued variational inequalities and fixed point problems, J. Fixed Point Theory Appl., 18 (2016), 27–43.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [9] H. Iiduka, W. Takahashi, Strong convergence theorems for nonexpansive nonself-mappings and inverse-strongly-monotone mappings, J. Convex Anal., 11 (2004), 69–79.
  • MathSciNet
  • Google Scholar
  • MATH


• [10] H. Iiduka, W. Takahashi, Strong convergence theorems for nonexpansive mappings and inverse-strongly monotone mappings, Nonlinear Anal., 61 (2005), 341–350.
  • View Article
  • Google Scholar


• [11] H. Iiduka, W. Takahashi , Weak convergence of a projection algorithm for variational inequalities in a Banach space, J. Math. Anal. Appl., 339 (2008), 668–679.
  • View Article
  • MathSciNet
  • Google Scholar


• [12] G. M. Korpelevič, An extragradient method for finding saddle points and for other problems, (Russian) Èkonom. i Mat. Metody, 12 (1976), 747–756.
  • MathSciNet
  • MATH


• [13] R. Kraikaew, S. Saejung, Strong convergence of the Halpern subgradient extragradient method for solving variational inequalities in Hilbert spaces, J. Optim. Theory Appl., 163 (2014), 399–412.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [14] J.-L. Lions, Optimal control of systems governed by partial differential equations, Translated from the French by S. K. Mitter, Die Grundlehren der mathematischenWissenschaften, Band 170 Springer-Verlag, New York-Berlin (1971)
  • Google Scholar


• [15] Y. Liu, Strong convergence theorem for relatively nonexpansive mapping and inverse-strongly-monotone mapping in a Banach space, Appl. Math. Mech. (English Ed.), 30 (2009), 925–932.
  • View Article
  • MathSciNet
  • Google Scholar


• [16] Y. Liu, Strong convergence of the Halpern subgradient extragradient method for solving variational inequalities in Banach spaces, J. Nonlinear Sci. Appl., 10 (2017), 395–409.
  • View Article
  • MathSciNet
  • Google Scholar


• [17] Y. V. Malitsky, V. V. Semenov , An extragradient algorithm for monotone variational inequalities, Translation of Kibernet. Sistem. Anal., 2014 (2014), 125–131, Cybernet. Systems Anal., 50 (2014), 271–277.
  • View Article
  • MathSciNet
  • Google Scholar


• [18] S.-Y. Matsushita, W. Takahashi, A strong convergence theorem for relatively nonexpansive mappings in a Banach space, J. Approx. Theory, 134 (2005), 257–266.
  • View Article
  • Google Scholar


• [19] N. Nadezhkina, W. Takahashi , Strong convergence theorem by a hybrid method for nonexpansive mappings and Lipschitzcontinuous monotone mappings, SIAM J. Optim., 16 (2006), 1230–1241.
  • View Article
  • Google Scholar


• [20] K. Nakajo, Strong convergence for gradient projection method and relatively nonexpansive mappings in Banach spaces , Appl. Math. Comput., 271 (2015), 251–258.
  • View Article
  • MathSciNet
  • Google Scholar


• [21] L. D. Popov, A modification of the Arrow-Hurwitz method of search for saddle points, (Russian) Mat. Zametki, 28 (1980), 777–784.
  • MathSciNet


• [22] Y. Takahashi, K. Hashimoto, M. Kato, On sharp uniform convexity, smoothness, and strong type, cotype inequalities, J. Nonlinear Convex Anal., 3 (2002), 267–281.
  • MathSciNet
  • Google Scholar
  • MATH


• [23] W. Takahashi, M. Toyoda, Weak convergence theorems for nonexpansive mappings and monotone mappings, J. Optim. Theory Appl., 118 (2003), 417–428.
  • View Article
  • Google Scholar


• [24] K.-K. Tan, H. K. Xu, Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process, J. Math. Anal. Appl., 178 (1993), 301–308.
  • View Article
  • MathSciNet
  • Google Scholar


• [25] A. R. Tufa, H. Zegeye, An algorithm for finding a common point of the solutions of fixed point and variational inequality problems in Banach spaces, Arab. J. Math. (Springer), 4 (2015), 199–213.
  • View Article
  • MathSciNet
  • Google Scholar