Regularized Newton Methods for Monotone Variational Inequalities with H\"older Continuous Jacobians

TL;DR

This paper introduces regularized Newton methods tailored for monotone variational inequalities with H"older continuous Jacobians, achieving improved iteration complexity based on the H"older parameter.

Contribution

It develops both parameter-dependent and universal regularized Newton methods with new iteration complexity bounds for solving these inequalities.

Findings

Achieves $oxed{ ext{O}( ext{}\epsilon^{-2/(2+ u)})}$ iteration complexity with known H"older parameter.

Proposes a universal method with $oxed{ ext{O}( ext{}\epsilon^{-4/(3(1+ u))})}$ complexity when the parameter is unknown.

Extends Newton-type methods to handle H"older continuous Jacobians effectively.

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.