  This paper can be seen as an attempt of rethinking the {\em Extra-Gradient
Philosophy} for solving Variational Inequality Problems. We show that the
properly defined {\em Reduced Gradients} can be used instead for finding
approximate solutions to Composite Variational Inequalities by the higher-order
schemes. Our methods are optimal since their performance is proportional to the
lower worst-case complexity bounds for corresponding problem classes. They
enjoy the provable hot-start capabilities even being applied to minimization
problems. The primal version of our schemes demonstrates a linear rate of
convergence under an appropriate uniform monotonicity assumption.
