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An efficient proximal-gradient-based method, called proximal extrapolatedgradient method, is designed for solving monotone variational inequality inHilbert space. The proposed method extends the acceptable range of parametersto obtain larger step sizes. The step size is predicted based a localinformation of the operator and corrected by linesearch procedures to satisfy avery weak condition, which is even weaker than the boundedness of sequencegenerated and always holds when the operator is the gradient of a convexfunction. We establish its convergence and ergodic convergence rate in theoryunder the larger range of parameters. Furthermore, we improve numericalefficiency by employing the proposed method with non-monotonic step size, andobtain the upper bound of the parameter relating to step size by an extremelysimple example. Related numerical experiments illustrate the improvements inefficiency from the larger step size.

Proximal extrapolated gradient methods with prediction and correction for monotone variational inequalities