Generalized Mirror Prox: Solving Variational Inequalities with Monotone Operator, Inexact Oracle, and Unknown Hölder Parameters

Pavel Dvurechensky, Alexander Gasnikov, Fedor Stonyakin, Alexander Titov

Published 2018-06-13Version 1

We introduce an inexact oracle model for variational inequalities (VI) with monotone operator, propose a numerical method which solves such VI's and analyze its convergence rate. As a particular case, we consider VI's with H\"older continuous operator and show that our algorithm is universal. This means that without knowing the H\"older parameter $\nu$ and H\"older constant $L_{\nu}$ it has the best possible complexity for this class of VI's, namely our algorithm has complexity $O\left( \inf_{\nu\in[0,1]}\left(\frac{L_{\nu}}{\varepsilon} \right)^{\frac{2}{1+\nu}}R^2 \right)$, where $R$ is the size of the feasible set and $\varepsilon$ is the desired accuracy of the solution. We also consider the case of VI's with strongly monotone operator and generalize our method for VI's with inexact oracle and our universal method for this class of problems. Finally, we show, how our method can be applied to convex-concave saddle-point problems with H\"older continuous partial subgradients.