Abstract
In this paper, we investigate the convergence properties of an iterative method for solving variational inequalities in Euclidean space. We show that under certain assumptions the method can be applied to variational inequalities defined over the common fixed point set of a given infinite family of cutter operators. The main step of our method consists in the computation of the metric projection onto a certain superhalf- space, which is constructed using the input data defining the problem. Moreover, in the case where the common fixed point set is a finite intersection of the fixed point sets of cutters for which Opial’s closedness principle holds, we show that the iterative method can be easily combined with either sequential, composition or convex combination type methods. We also discuss ways of applying our method to more general classes of operators such as quasi-nonexpansive and demi-contractive ones.
Similar content being viewed by others
References
K. Aoyama, F. Kohsaka, Viscosity approximation process for a sequence of quasinonexpansive mappings. Fixed Point Theory Appl. 2014 (2014), doi:10.1186/1687-1812-2014-17, 11 pages
Bauschke H.H.: The approximation of fixed points of composition of nonexpansive mappings in Hilbert space. J. Math. Anal. Appl. 202, 150–159 (1996)
Bauschke H.H., Borwein J.: On the convergence of von Neumann’s alternating projection algorithm for two sets. Set-Valued Anal. 1, 185–212 (1993)
Bauschke H.H., Borwein J.: On projection algorithms for solving convex feasibility problems. SIAM Rev. 38, 367–426 (1996)
Bauschke H.H., Combettes P.L.: A weak-to-strong convergence principle for Fejér-monotone methods in Hilbert spaces. Math. Oper. Res. 26, 248–264 (2001)
Bauschke H.H., Combettes P.L.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces. Springer, New York (2011)
Bauschke H.H., Matoušková E., Reich S.: Projection and proximal point methods: Convergence results and counterexamples. Nonlinear Anal. 56, 715–738 (2004)
Bauschke H.H., Wang C., Wang X., Xu J.: On subgradient projectors. SIAM J. Optim. 25, 1064–1082 (2015)
Bruck R.E.: Random products of contractions in metric and Banach spaces. J. Math. Anal. Appl. 8, 319–332 (1982)
A. Cegielski, Generalized relaxations of nonexpansive operators and convex feasibility problems. In: Nonlinear Analysis and Optimization. I. Nonlinear Analysis, Contemp. Math. 513, Amer. Math. Soc., Providence, RI, 2010, 111–123
A. Cegielski, Iterative Methods for Fixed Point Problems in Hilbert Spaces. Springer, Heidelberg, 2012
Cegielski A.: Extrapolated simultaneous subgradient projection method for variational inequality over the intersection of convex subsets. J. Nonlinear Convex Anal. 15, 211–218 (2014)
A. Cegielski, Application of quasi-nonexpansive operators to an iterative method for variational inequality. In preparation
A. Cegielski and Y. Censor, Opial-type theorems and the common fixed point problem. In: Fixed-Point Algorithms for Inverse Problems in Science and Engineering, Springer Optim. Appl. 49, Springer, New York, 2011, 155–183
Cegielski A., Gibali A., Reich S., Zalas R.: An algorithm for solving the variational inequality problem over the fixed point set of a quasi-nonexpansive operator in Euclidean space. Numer. Funct. Anal. Optim. 34, 1067–1096 (2013)
Cegielski A., Zalas R.: Methods for variational inequality problem over the intersection of fixed point sets of quasi-nonexpansive operators. Numer. Funct. Anal. Optim. 34, 255–283 (2013)
Cegielski A., Zalas R.: Properties of a class of approximately shrinking operators and their applications. Fixed Point Theory 15, 399–426 (2014)
Y. Censor and A. Gibali, Projections onto super-half-spaces for monotone variational inequality problems in finite-dimensional space. J. Nonlinear Convex Anal. 9, 461–75 (2008)
Censor Y., Segal A.: On the string averaging method for sparse common fixed point problems. Int. Trans. Oper. Res. 16, 481–494 (2009)
Y. Censor and A. Segal, Sparse string-averaging and split common fixed points. In: Nonlinear Analysis and Optimization. I. Nonlinear Analysis, Contemp. Math. 513, Amer. Math. Soc., Providence, RI, 2010, 125–142
P. L. Combettes, Quasi-Fejérian analysis of some optimization algorithms. In: Inherently Parallel Algorithms in Feasibility and Optimization and Their Applications (Haifa, 2000), Stud. Comput. Math. 8, North-Holland, Amsterdam, 2001, 115–152
Crombez G.: A hierarchical presentation of operators with fixed points on Hilbert spaces. Numer. Funct. Anal. Optim. 27, 259–277 (2006)
Deutsch F.: Best Approximation in Inner Product Spaces. Springer, New York (2001)
Deutsch F., Yamada I.: Minimizing certain convex functions over the intersection of the fixed point sets of nonexpansive mappings. Numer. Funct. Anal. Optim. 19, 33–56 (1998)
Dotson W.G. Jr.: On the Mann iterative process. Trans. Amer. Math. Soc. 149, 65–73 (1970)
F. Facchinei and J.-S. Pang, Finite-Dimensional Variational Inequalities and Complementarity Problems. Volumes I and II, Springer, New York, 2003
Fletcher R.: Practical Methods of Optimization. John Wiley, Chichester (1987)
Fukushima M.: A relaxed projection method for variational inequalities. Math. Program. 35, 58–70 (1986)
Genel A., Lindenstrauss J.: An example concerning fixed points. Israel J. Math. 22, 81–86 (1975)
Goldstein A.A.: Convex programming in Hilbert space. Bull. Amer. Math. Soc. 70, 709–710 (1964)
Halperin I.: The product of projection operators. Acta Sci. Math. (Szeged) 23, 96–99 (1962)
Hirstoaga S.A.: Iterative selection methods for common fixed point problems. J. Math. Anal. Appl. 324, 1020–1035 (2006)
Kinderlehrer D., Stampacchia G.: An Introduction to Variational Inequalities and Their Applications. Academic Press, New York (1980)
Levitin E.S., Polyak B.T.: Constrained minimization methods. Zh. Vychisl. Mat. Mat. Fiz. 6, 787–823 (1966)
Măruşter Şt.: The solution by iteration of nonlinear equations in Hilbert spaces. Proc. Amer. Math. Soc. 63, 69–73 (1977)
Măruşter Şt.: Quasi-nonexpansivity and the convex feasibility problem. An. Ştiinţ. Univ. Al. I. Cuza Iaşi Inform. (N.S.) 15, 47–56 (2005)
Măruşter Şt., Popirlan C.: On the Mann-type iteration and the convex feasibility problem. J. Comput. Appl. Math. 212, 390–396 (2008)
Noor M.A.: Some developments in general variational inequalities. Appl. Math. Comput. 152, 199–277 (2004)
Opial Z.: Weak convergence of the sequence of successive approximations for nonexpansive mappings. Bull. Amer. Math. Soc. 73, 591–597 (1967)
Reich S., Zaslavski A.J.: Attracting mappings in Banach and hyperbolic spaces. J. Math. Anal. Appl. 253, 250–268 (2001)
Schott D.: Ball intersection model for Fejér zones of convex closed sets. Discuss. Math. Differ. Incl. Control Optim. 21, 51–79 (2001)
A. Segal, Directed operators for common fixed point problems and convex programming problems. PhD thesis, University of Haifa, Haifa, Israel, 2008
Song Y.: On a Mann type implicit iteration process for continuous pseudocontractive mappings. Nonlinear Anal. 67, 3058–3063 (2007)
Takahashi W., Takeuchi Y., Kubota R.: Strong convergence theorems by hybrid methods for families of nonexpansive mappings in Hilbert spaces. J. Math. Anal. Appl. 341, 276–286 (2008)
Tricomi F.: Un teorema sulla convergenza delle successioni formate delle successive iterate di una funzione di una variabile reale. Giorn. Mat. Battaglini 54, 1–9 (1916)
V. V. Vasin and A. L. Ageev, Ill-Posed Problems with a priori Information. VSP, Utrecht, 1995
Xiu N., Zhang J.: Some recent advances in projection-type methods for variational inequalities. J. Comput. Appl. Math. 152, 559–585 (2003)
Xu H.-K., Kim T.H.: Convergence of hybrid steepest-descent methods for variational inequalities. J. Optim. Theory Appl. 119, 185–201 (2003)
I. Yamada, The hybrid steepest descent method for the variational inequality problem over the intersection of fixed point sets of nonexpansive mappings. In: Inherently Parallel Algorithms in Feasibility and Optimization and Their Applications (Haifa, 2000), Stud. Comput. Math. 8, North-Holland, Amsterdam, 2001, 473–504
Yamada I., Ogura N.: Hybrid steepest descent method for variational inequality problem over the fixed point set of certain quasi-nonexpansive mappings. Numer. Funct. Anal. Optim. 25, 619–655 (2004)
I. Yamada, M. Yukawa and M. Yamagishi, Minimizing the Moreau envelope of nonsmooth convex functions over the fixed point set of certain quasinonexpansive mappings. In: Fixed-Point Algorithms for Inverse Problems in Science and Engineering, Springer Optim. Appl. 49, Springer, New York, 2011, 345–390
M. Zaknoon, Algorithmic developments for the convex feasibility problem. PhD thesis, University of Haifa, Haifa, Israel, 2003
R. Zalas, Variational inequalities for fixed point problems of quasi-nonexpansive operators. PhD thesis, University of Zielona Góra, Zielona Góra, Poland, 2014 (in Polish).
Zeidler E.: Nonlinear Functional Analysis and Its Applications. III. Variational Methods and Optimization. Springer, New York (1985)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Gibali, A., Reich, S. & Zalas, R. Iterative methods for solving variational inequalities in Euclidean space. J. Fixed Point Theory Appl. 17, 775–811 (2015). https://doi.org/10.1007/s11784-015-0256-x
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11784-015-0256-x