A new modified extra-gradient method for variational inequality problems (Q2746476)

The classical variational inequality problem to find a vector \(u^*\in\Omega\) such that NEWLINE\[NEWLINE\langle u- u^*, F(y^*)\rangle\geq 0\quad\text{for all }u\in \Omega,NEWLINE\]NEWLINE where \(\Omega\) is a nonempty closed convex subset of \(\mathbb{R}^n\), and \(F\) is a mapping from \(\mathbb{R}^n\) into itself is considered. For solving this problem the new modified extra-gradient method is proposed. It is almost as simple as the original extragradient method which contains only two projection at each iteration, and the convergence of the method is proved under a mild condition that the underlying mapping \(F\) is continuous and satisfies a generalized monotonicity. Preliminary computational results are given to illustrate the efficiency of the proposed method.