Regularized Newton Methods for Monotone Variational Inequalities with Hölder Continuous Jacobians

Abstract: This paper considers the problems of solving monotone variational inequalities with H\"older continuous Jacobians. By employing the knowledge of H\"older parameter $\nu$, we propose the $\nu$-regularized extra-Newton method within at most $\mathcal O(\epsilon^{-2/(2+\nu)})$ iterations to obtain an $\epsilon$-accurate solution. In the case of $\nu$ is unknown, we propose the universal regularized extra-Newton method within $\mathcal O(\epsilon^{-4/(3(1+\nu))})$ iteration complexity.