**Keywords:**

In this paper, we generalize the classical extragradient algorithm for solving variational inequality problems

by utilizing non-null normal vectors of the feasible set. In particular, two conceptual algorithms are proposed and each of them has three different variants which are related to modified extragradient algorithms.

Our analysis contains two main parts: The first part

contains two different linesearches, one on the boundary of the

feasible set and the other one along the feasible direction. The

linesearches allow us to find suitable halfspaces containing the

solution set of the problem. By using non-null normal vectors of the

feasible set, these linesearches can potentially accelerate the

convergence. If all normal vectors are chosen as zero, then some of

these variants reduce to several well-known projection methods

proposed in the literature for solving the variational inequality

problem. The second part consists of three special projection steps,

generating three sequences with different interesting features.

Convergence analysis of both conceptual algorithms is

established assuming existence of solutions, continuity and a weaker

condition than pseudomonotonicity on the operator. Examples, on each

variant, show that the modifications proposed here perform better

than previous classical variants. These results suggest that our

scheme may significantly improve the extragradient variants.