Regularized Newton Methods for Monotone Variational Inequalities with Hölder Continuous Jacobians

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