# Gradient Methods for Problems with Inexact Model of the Objective

## TL;DR

This paper develops gradient methods for convex optimization problems with inexact objective models, providing convergence analysis and applying the framework to clustering, optimal transport, and barycenter problems.

## Contribution

It introduces a general inexact model framework encompassing various inexact oracle conditions and analyzes gradient methods within this setting.

## Key findings

- Convergence rates are established for convex and strongly convex cases.
- Proximal Sinkhorn algorithm effectively approximates optimal transport distance.
- Proximal Iterative Bregman Projections algorithm efficiently computes optimal transport barycenters.

## Abstract

We consider optimization methods for convex minimization problems under inexact information on the objective function. We introduce inexact model of the objective, which as a particular cases includes $(\delta,L)$ inexact oracle and relative smoothness condition. We analyze gradient method which uses this inexact model and obtain convergence rates for convex and strongly convex problems. To show potential applications of our general framework we consider three particular problems. The first one is clustering by electorial model introduced in [Nesterov, 2018]. The second one is approximating optimal transport distance, for which we propose a Proximal Sinkhorn algorithm. The third one is devoted to approximating optimal transport barycenter and we propose a Proximal Iterative Bregman Projections algorithm. We also illustrate the practical performance of our algorithms by numerical experiments.