# Inexact Model: A Framework for Optimization and Variational Inequalities

## TL;DR

This paper introduces a versatile framework for first-order optimization and variational inequalities using inexact models, unifying existing methods and enabling the development of new algorithms with optimal complexity.

## Contribution

The paper presents a general inexact model framework that unifies many optimization methods and introduces a universal method for variational inequalities with optimal complexity.

## Key findings

- Reproduces known optimization algorithms as special cases
- Develops a universal method for variational inequalities
- Achieves optimal complexity without prior smoothness knowledge

## Abstract

In this paper we propose a general algorithmic framework for first-order methods in optimization in a broad sense, including minimization problems, saddle-point problems and variational inequalities. This framework allows to obtain many known methods as a special case, the list including accelerated gradient method, composite optimization methods, level-set methods, proximal methods. The idea of the framework is based on constructing an inexact model of the main problem component, i.e. objective function in optimization or operator in variational inequalities. Besides reproducing known results, our framework allows to construct new methods, which we illustrate by constructing a universal method for variational inequalities with composite structure. This method works for smooth and non-smooth problems with optimal complexity without a priori knowledge of the problem smoothness. We also generalize our framework for strongly convex objectives and strongly monotone variational inequalities.