Extragradient Methods and Linesearch Algorithms for Solving Ky Fan Inequalities and Fixed Point Problems

In this paper, we introduce some new iterative methods for finding a common element of the set of points satisfying a Ky Fan inequality, and the set of fixed points of a contraction mapping in a Hilbert space. The strong convergence of the iterates generated by each method is obtained thanks to a hybrid projection method, under the assumptions that the fixed-point mapping is a ξ-strict pseudocontraction, and the function associated with the Ky Fan inequality is pseudomonotone and weakly continuous. A Lipschitz-type condition is assumed to hold on this function when the basic iteration comes from the extragradient method. This assumption is unnecessary when an Armijo backtracking linesearch is incorporated in the extragradient method. The particular case of variational inequality problems is examined in a last section.

[1]  G. M. Korpelevich.  The extragradient method for finding saddle points and other problems , 1976 .

[2]  Heinz H. Bauschke,et al.  A Weak-to-Strong Convergence Principle for Fejé-Monotone Methods in Hilbert Spaces , 2001, Math. Oper. Res..

[3]  Wataru Takahashi,et al.  Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces , 2007 .

[4]  Nicolas Hadjisavvas,et al.  Strong convergence theorem by a hybrid extragradient-like approximation method for variational inequalities and fixed point problems , 2010, J. Glob. Optim..

[5]  Isao Yamada,et al.  A Use of Conjugate Gradient Direction for the Convex Optimization Problem over the Fixed Point Set of a Nonexpansive Mapping , 2008, SIAM J. Optim..

[6]  J. Borwein,et al.  Epigraphical and Uniform Convergence of Convex Functions , 1996 .

[7]  I. Ekeland,et al.  Convex analysis and variational problems , 1976 .

[8]  Jen-Chih Yao,et al.  An extragradient iterative scheme by viscosity approximation methods for fixed point problems and variational inequality problems , 2009 .

[9]  W. R. Mann,et al.  Mean value methods in iteration , 1953 .

[10]  Paul-Emile Maingé,et al.  A Hybrid Extragradient-Viscosity Method for Monotone Operators and Fixed Point Problems , 2008, SIAM J. Control. Optim..

[11]  Jen-Chih Yao,et al.  Hybrid Proximal-Type and Hybrid Shrinking Projection Algorithms for Equilibrium Problems, Maximal Monotone Operators, and Relatively Nonexpansive Mappings , 2010 .

[12]  Le Dung Muu,et al.  Dual extragradient algorithms extended to equilibrium problems , 2011, Journal of Global Optimization.

[13]  Poom Kumam,et al.  Strong convergence theorems for solving equilibrium problems and fixed point problems of ξ-strict pseudo-contraction mappings by two hybrid projection methods , 2010, J. Comput. Appl. Math..

[14]  Jen-Chih Yao,et al.  On modified iterative method for nonexpansive mappings and monotone mappings , 2007, Appl. Math. Comput..

[15]  Y. Shehu.  Fixed point solutions of variational inequality and generalized equilibrium problems with applications , 2010 .

[16]  L. Muu,et al.  Convergence of an adaptive penalty scheme for finding constrained equilibria , 1992 .

[17]  W. Takahashi,et al.  Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups , 2003 .

[18]  P. Ánh.  A hybrid extragradient method extended to fixed point problems and equilibrium problems , 2013 .

[19]  G. Mastroeni.  On Auxiliary Principle for Equilibrium Problems , 2003 .

[20]  Jean-Jacques Strodiot,et al.  A bundle method for solving equilibrium problems , 2009, Math. Program..

[21]  Poom Kumam,et al.  A new hybrid iterative method for solution of equilibrium problems and fixed point problems for an inverse strongly monotone operator and a nonexpansive mapping , 2009 .

[22]  Wataru Takahashi,et al.  Weak and Strong Convergence Theorems for a Nonexpansive Mapping and an Equilibrium Problem , 2007 .

[23]  Jen-Chih Yao,et al.  An extragradient-like approximation method for variational inequality problems and fixed point problems , 2007, Appl. Math. Comput..

[24]  W. Oettli,et al.  From optimization and variational inequalities to equilibrium problems , 1994 .

[25]  Antonino Maugeri,et al.  Equilibrium problems and variational models , 2003 .

[26]  Isao Yamada,et al.  A subgradient-type method for the equilibrium problem over the fixed point set and its applications , 2009 .

[27]  Shin Min Kang,et al.  An extragradient-type method for generalized equilibrium problems involving strictly pseudocontractive mappings , 2011, J. Glob. Optim..

[28]  Hideaki Iiduka,et al.  A new iterative algorithm for the variational inequality problem over the fixed point set of a firmly nonexpansive mapping , 2010 .

[29]  Wataru Takahashi,et al.  Weak Convergence Theorem by an Extragradient Method for Nonexpansive Mappings and Monotone Mappings , 2006 .

[30]  T. D. Quoc,et al.  Extragradient algorithms extended to equilibrium problems , 2008 .

[31]  A. Moudafi.  Viscosity Approximation Methods for Fixed-Points Problems , 2000 .

[32]  C. Zălinescu.  Convex analysis in general vector spaces , 2002 .

[33]  Wataru Takahashi,et al.  Strong Convergence Theorem by a Hybrid Method for Nonexpansive Mappings and Lipschitz-Continuous Monotone Mappings , 2006, SIAM J. Optim..

[34]  Lu-Chuan Ceng,et al.  MODIFIED EXTRAGRADIENT METHODS FOR STRICT PSEUDO-CONTRACTIONS AND MONOTONE MAPPINGS , 2009 .

[35]  Phayap Katchang,et al.  A viscosity of extragradient approximation method for finding equilibrium problems, variational inequalities and fixed point problems for nonexpansive mappings , 2009 .

[36]  Paul-Emile Maingé,et al.  Projected subgradient techniques and viscosity methods for optimization with variational inequality constraints , 2010, Eur. J. Oper. Res..