Strong convergence result for monotone variational inequalities

Our aim in this paper is to study strong convergence results for L-Lipschitz continuous monotone variational inequality but L is unknown using a combination of subgradient extra-gradient method and viscosity approximation method with adoption of Armijo-like step size rule in infinite dimensional real Hilbert spaces. Our results are obtained under mild conditions on the iterative parameters. We apply our result to nonlinear Hammerstein integral equations and finally provide some numerical experiments to illustrate our proposed algorithm.

[1]  Paul-Emile Maingé,et al.  Convergence of One-Step Projected Gradient Methods for Variational Inequalities , 2016, J. Optim. Theory Appl..

[2]  J. Aubin,et al.  Applied Nonlinear Analysis , 1984 .

[3]  An existence theorem for Hammerstein integral equations. , 1994 .

[4]  Józef Banaś,et al.  Integrable solutions of Hammerstein and Urysohn integral equations , 1989, Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics.

[5]  Alfredo N. Iusem,et al.  Korpelevich’s method for variational inequality problems in Banach spaces , 2011, J. Glob. Optim..

[6]  Haim Brezis,et al.  Existence theorems for nonlinear integral equations of Hammerstein type , 1975 .

[7]  C. Chidume,et al.  Solution of nonlinear integral equations of Hammerstein type , 2011 .

[8]  C. P. Gupta,et al.  Maximal monotone operators and nonlinear integral equations of Hammerstein type , 1970 .

[9]  Paul-Emile Maingé.  Numerical approach to monotone variational inequalities by a one-step projected reflected gradient method with line-search procedure , 2016, Comput. Math. Appl..

[10]  Yekini Shehu,et al.  Iterative approximation of solutions of equations of Hammerstein type in certain Banach spaces , 2013, Appl. Math. Comput..

[11]  Y. Shehu,et al.  Strong convergence theorem for approximation of solutions of equations of Hammerstein type , 2012 .

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

[13]  V. Berinde.  Iterative Approximation of Fixed Points , 2007 .

[14]  C. P. Gupta,et al.  On the Variational Method for the Existence of Solutions of Nonlinear Equations of Hammerstein Type , 1973 .

[15]  Felix E. Browder,et al.  Nonlinear mappings of nonexpansive and accretive type in Banach spaces , 1967 .

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

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

[18]  A. Hammerstein.  Nichtlineare Integralgleichungen nebst Anwendungen , 1930 .

[19]  Yekini Shehu,et al.  Convergence theorems for maximal monotone operators and fixed point problems in Banach spaces , 2014, Appl. Math. Comput..

[20]  Yu. V. Malitsky,et al.  A hybrid method without extrapolation step for solving variational inequality problems , 2015, J. Glob. Optim..

[21]  H. Zegeye,et al.  Approximation of solutions of nonlinear equations of Hammerstein type in Hilbert space , 2004 .

[22]  Józef Banaś,et al.  Measures of weak noncompactness and nonlinear integral equations of convolution type , 1990 .

[23]  A. Cegielski.  Iterative Methods for Fixed Point Problems in Hilbert Spaces , 2012 .

[24]  M. Solodov,et al.  A New Projection Method for Variational Inequality Problems , 1999 .

[25]  Paul Tseng,et al.  A Modified Forward-backward Splitting Method for Maximal Monotone Mappings 1 , 1998 .

[26]  A. Iusem,et al.  A variant of korpelevich’s method for variational inequalities with a new search strategy , 1997 .

[27]  Changjie Fang,et al.  Some Extragradient Algorithms for Variational Inequalities , 2015 .

[28]  Heinz H. Bauschke,et al.  Convex Analysis and Monotone Operator Theory in Hilbert Spaces , 2011, CMS Books in Mathematics.

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

[30]  Maximal monotone operators in nonreflexive Banach spaces and nonlinear integral equations of Hammerstein type , 1975 .

[31]  K. Latrach,et al.  Existence results for a generalized nonlinear Hammerstein equation on L1 spaces , 2007 .

[32]  Satit Saejung,et al.  Strong Convergence of the Halpern Subgradient Extragradient Method for Solving Variational Inequalities in Hilbert Spaces , 2014, J. Optim. Theory Appl..

[33]  G. Emmanuele,et al.  End of CMS Special Session Papers Integrable Solutions of a Functional-Integral Equation , 1992 .

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

[35]  C. Baiocchi,et al.  Variational and quasivariational inequalities: Applications to free boundary problems , 1983 .

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

[37]  Yekini Shehu,et al.  Strong convergence theorem for integral equations of Hammerstein type in Hilbert spaces , 2014, Appl. Math. Comput..

[38]  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..

[39]  I. Konnov.  Combined Relaxation Methods for Variational Inequalities , 2000 .

[40]  F. Browder,et al.  Some new results about Hammerstein equations , 1974 .

[41]  Xiaoming Yuan,et al.  An approximate proximal-extragradient type method for monotone variational inequalities , 2004 .

[42]  R. Glowinski,et al.  Numerical Analysis of Variational Inequalities , 1981 .

[43]  Abdellah Bnouhachem.  A self-adaptive method for solving general mixed variational inequalities , 2005 .

[44]  F. Facchinei,et al.  Finite-Dimensional Variational Inequalities and Complementarity Problems , 2003 .

[45]  Hong-Kun Xu.  Iterative Algorithms for Nonlinear Operators , 2002 .

[46]  W. Hager.  Review: R. Glowinski, J. L. Lions and R. Trémolières, Numerical analysis of variational inequalities , 1983 .

[47]  S. I. Lyashko,et al.  Low-cost modification of Korpelevich’s methods for monotone equilibrium problems , 2011 .

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

[49]  A new non-Lipschitzian projection method for solving variational inequalities in Euclidean spaces , 2015 .

[50]  C. P. Gupta,et al.  Monotone operators and nonlinear integral equations of Hammerstein type , 1969 .

[51]  A. Nagurney.  Network Economics: A Variational Inequality Approach , 1992 .

[52]  D. Kinderlehrer,et al.  An introduction to variational inequalities and their applications , 1980 .

[53]  Józef Banas,et al.  Integrable solutions of a functional-integral equation. , 1989 .

[54]  Siu-Ah Ng,et al.  Donsker's Delta Function and the Covariance between Generalized Functionals , 2002 .

[55]  E. Khobotov.  Modification of the extra-gradient method for solving variational inequalities and certain optimization problems , 1989 .