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The variational inequality problem in finite-dimensional Euclidean space isaddressed in this paper, and two inexact variants of the extragradient methodare proposed to solve it. Instead of computing exact projections on theconstraint set, as in previous versions extragradient method, the proposedmethods compute feasible inexact projections on the constraint set using arelative error criterion. The first version of the proposed method provided isa counterpart to the classic form of the extragradient method with constantsteps. In order to establish its convergence we need to assume that theoperator is pseudo-monotone and Lipschitz continuous, as in the standardapproach. For the second version, instead of a fixed step size, the methodpresented finds a suitable step size in each iteration by performing a linesearch. Like the classical extragradient method, the proposed method does justtwo projections into the feasible set in each iteration. A full convergenceanalysis is provided, with no Lipschitz continuity assumption of the operatordefining the variational inequality problem.

Extragradient method with feasible inexact projection to variational inequality problem