Development of a method for solving structural optimization problems

TL;DR

This paper introduces a unified approach to gradient-based optimization methods leveraging the concept of inexact models, enabling faster convergence for structured problems like min-max, transportation, and clustering.

Contribution

It proposes a unification of gradient methods through inexact models, creating adaptable algorithms for various structured optimization problems.

Findings

Developed gradient methods for functions with relative smoothness

Created primal-dual adaptive and fast gradient methods

Supported inexact models with diverse optimization problem examples

Abstract

In practice, optimization tasks have some structure that allows developing new algorithms for every problem with faster convergence rates. Using the structure of optimization tasks, we can propose algorithms with more optimistic convergence rates for the following optimization problems: functions with Holder continuous gradients, superposition of functions (min-max problems), transportation problems, clustering by electorial model. In this work, we propose the unification of gradient-type methods into one method using a special concept of inexact model and develop a series of methods that can solve generalized optimization problem statements and use its structure with the aid of the proposed concept of inexact model. We constructed the gradient method for problems with relative smoothness, the primal--dual adaptive gradient and fast gradient methods, and the stochastic nonadaptive gradient methods that support an inexact model of a function. Moreover, the concept of inexact model is supported by different examples of optimization problems.