We present a subgradient extragradient method for solving variational inequalities in Hilbert space. In addition, we propose a modified version of our algorithm that finds a solution of a variational inequality which is also a fixed point of a given nonexpansive mapping. We establish weak convergence theorems for both algorithms.
We propose a prototypical Split Inverse Problem (SIP) and a new variational
problem, called the Split Variational Inequality Problem (SVIP), which is a
SIP. It entails finding a solution of one inverse problem (e.g., a Variational
Inequality Problem (VIP)), the image of which under a given bounded linear
transformation is a solution of another inverse problem such as a VIP. We
construct iterative algorithms that solve such problems, under reasonable
conditions, in Hilbert space and then discuss special cases, some of which are
new even in Euclidean space.Comment: This is a revised version of arXiv:1009.3780v1. Accepted for
publication in the journal Numerical Algorithm
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