High-Order Reduced-Gradient Methods for Composite Variational Inequalities

This paper can be seen as an attempt of rethinking the Extra-Gradient Philosophy for solving Variational Inequality Problems. We show that the properly defined 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.