Retraction algorithms for solving variational inequalities, pseudomonotone equilibrium problems, and fixed-point problems in Banach spaces

In this paper, using sunny generalized nonexpansive retractions which are different from the metric projection and generalized metric projection in Banach spaces, we present new extragradient and line search algorithms for finding the solution of a J-variational inequality whose constraint set is the common elements of the set of fixed points of a family of generalized nonexpansive mappings and the set of solutions of a pseudomonotone J-equilibrium problem for a J -α-inverse-strongly monotone operator in a Banach space. To prove strong convergence of generated iterates in the extragradient method, we introduce a ϕ∗-Lipschitz-type condition and assume that the equilibrium bifunction satisfies this condition. This condition is unnecessary when the line search method is used instead of the extragradient method. Using FMINCON optimization toolbox in MATLAB, we give some numerical examples and compare them with several existence results in literature to illustrate the usability of our results.

[1]  Jen-Chih Yao,et al.  Nonlinear Ergodic Theorem for Positively Homogeneous Nonexpansive Mappings in Banach Spaces , 2014 .

[2]  Zeynab Jouymandi,et al.  Extragradient Methods for Solving Equilibrium Problems, Variational Inequalities, and Fixed Point Problems , 2017 .

[3]  G. Cohen.  Auxiliary problem principle and decomposition of optimization problems , 1980 .

[4]  H. Zegeye.  Strong convergence theorems for maximal monotone mappings in Banach spaces , 2008 .

[5]  Yair Censor,et al.  The Subgradient Extragradient Method for Solving Variational Inequalities in Hilbert Space , 2011, J. Optim. Theory Appl..

[6]  J. Aubin.  Optima and Equilibria , 1993 .

[7]  W. Takahashi.  Nonlinear Functional Analysis , 2000 .

[8]  Shangquan Bu,et al.  WEAK CONVERGENCE THEOREMS FOR GENERAL EQUILIBRIUM PROBLEMS AND VARIATIONAL INEQUALITY PROBLEMS AND FIXED POINT PROBLEMS IN BANACH SPACES , 2013 .

[9]  Anatoly Antipin.  The convergence of proximal methods to fixed points of extremal mappings and estimates of their rate of convergence , 1995 .

[10]  S. Reich,et al.  Asymptotic Behavior of Relatively Nonexpansive Operators in Banach Spaces , 2001 .

[11]  Fixed point results for multivalued contractions on ordered gauge spaces , 2009 .

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

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

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

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

[16]  Habtu Zegeye,et al.  Strong convergence theorems for variational inequality problems and quasi-$${\phi}$$-asymptotically nonexpansive mappings , 2012, J. Glob. Optim..

[17]  Ravi P. Agarwal,et al.  Fixed Point Theory for Lipschitzian-type Mappings with Applications , 2009 .

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

[19]  L. D. Muu,et al.  Regularization Algorithms for Solving Monotone Ky Fan Inequalities with Application to a Nash-Cournot Equilibrium Model , 2009, J. Optimization Theory and Applications.

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

[21]  S. Reich.  Book Review: Geometry of Banach spaces, duality mappings and nonlinear problems , 1992 .

[22]  A. Dax.  The smallest point of a polytope , 1990 .

[23]  C. E. Chidume,et al.  Geometric Properties of Banach Spaces and Nonlinear Iterations , 2009 .

[24]  G. Cohen.  Auxiliary problem principle extended to variational inequalities , 1988 .

[25]  Phan Tu Vuong,et al.  Extragradient Methods and Linesearch Algorithms for Solving Ky Fan Inequalities and Fixed Point Problems , 2012, J. Optim. Theory Appl..

[26]  A Strong Convergence Theorem for Equilibrium problems and Generalized Hybrid mappings , 2016 .

[27]  Zeynab Jouymandi,et al.  Extragradient and linesearch algorithms for solving equilibrium problems, variational inequalities and fixed point problems in Banach spaces , 2019, Fixed Point Theory.

[28]  J. Strodiot,et al.  On extragradient-viscosity methods for solving equilibrium and fixed point problems in a Hilbert space , 2015 .

[29]  Simeon Reich,et al.  Asymptotic behavior of contractions in Banach spaces , 1973 .

[30]  I. Ciorǎnescu.  Geometry of banach spaces, duality mappings, and nonlinear problems , 1990 .

[31]  S. Reich,et al.  Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings , 1984 .

[32]  Yair Censor,et al.  Extensions of Korpelevich's extragradient method for the variational inequality problem in Euclidean space , 2012 .

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

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

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

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

[37]  Yair Censor,et al.  Strong convergence of subgradient extragradient methods for the variational inequality problem in Hilbert space , 2011, Optim. Methods Softw..

[38]  W. Takahashi,et al.  STRONG CONVERGENCE THEOREMS BY MONOTONE HYBRID METHOD FOR A FAMILY OF GENERALIZED NONEXPANSIVE MAPPINGS IN BANACH SPACES , 2012 .

[39]  M. Noor.  Extragradient Methods for Pseudomonotone Variational Inequalities , 2003 .

[40]  Wataru Takahashi,et al.  Strong Convergence of a Proximal-Type Algorithm in a Banach Space , 2002, SIAM J. Optim..

[41]  Wataru Takahashi,et al.  A new projection and convergence theorems for the projections in Banach spaces , 2007, J. Approx. Theory.

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

[43]  Nenad Ujević,et al.  An iterative method for solving nonlinear equations , 2007 .

[44]  S. Reich,et al.  Nonexpansive Retracts in Banach Spaces , 2007 .

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

[46]  Shoham Sabach,et al.  Iterative Methods for Solving Systems of Variational Inequalities in Reflexive Banach Spaces , 2011, SIAM J. Optim..

[47]  Jean Jacques Strodiot,et al.  A new class of hybrid extragradient algorithms for solving quasi-equilibrium problems , 2013, J. Glob. Optim..