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In this paper we introduce a new self-adaptive iterative algorithm for solving the variational inequalities in real Hilbert spaces, denoted by    \documentclass[12pt]{minimal} 				\usepackage{amsmath} 				\usepackage{wasysym}  				\usepackage{amsfonts}  				\usepackage{amssymb}  				\usepackage{amsbsy} 				\usepackage{mathrsfs} 				\usepackage{upgreek} 				\setlength{\oddsidemargin}{-69pt} 				\begin{document}$\operatorname{VI}(C, F)$\end{document}   VI  (  C  ,  F  ). Here    \documentclass[12pt]{minimal} 				\usepackage{amsmath} 				\usepackage{wasysym}  				\usepackage{amsfonts}  				\usepackage{amssymb}  				\usepackage{amsbsy} 				\usepackage{mathrsfs} 				\usepackage{upgreek} 				\setlength{\oddsidemargin}{-69pt} 				\begin{document}$C\subseteq \mathcal{H}$\end{document}   C  ⊆  His a nonempty, closed and convex set and    \documentclass[12pt]{minimal} 				\usepackage{amsmath} 				\usepackage{wasysym}  				\usepackage{amsfonts}  				\usepackage{amssymb}  				\usepackage{amsbsy} 				\usepackage{mathrsfs} 				\usepackage{upgreek} 				\setlength{\oddsidemargin}{-69pt} 				\begin{document}$F: C\rightarrow \mathcal{H}$\end{document}   F  :  C  →  His boundedly Lipschitz continuous (i.e., Lipschitz continuous on any bounded subset of  C ) and strongly monotone operator. One of the advantages of our algorithm is that it does not require the knowledge of the Lipschitz constant of  F  on any bounded subset of  C  or the strong monotonicity coefficient a priori. Moreover, the proposed self-adaptive step size rule only adds a small amount of computational effort and hence guarantees fast convergence rate. Strong convergence of the method is proved and a posteriori error estimate of the convergence rate is obtained.  Primary numerical results illustrate the behavior of our proposed scheme and also suggest that the convergence rate of the method is comparable with the classical gradient projection method for solving variational inequalities.

Self-adaptive iterative method for solving boundedly Lipschitz continuous and strongly monotone variational inequalities