Composite Optimization using Local Models and Global Approximations

TL;DR

This paper introduces a unified framework combining global approximations with local models to effectively solve complex nonconvex and nonsmooth composite optimization problems, enhancing convergence and algorithmic flexibility.

Contribution

It develops a novel double-loop algorithmic framework that integrates local models with global approximations for challenging composite optimization tasks.

Findings

Convergence of near-stationary points to original problem stationary points

Development of practical algorithms using convex master programs

Framework applicable to a wide class of nonconvex, nonsmooth problems

Abstract

This work presents a unified framework that combines global approximations with locally built models to handle challenging nonconvex and nonsmooth composite optimization problems, including cases involving extended real-valued functions. We show that near-stationary points of the approximating problems converge to stationary points of the original problem under suitable conditions. Building on this, we develop practical algorithms that use tractable convex master programs derived from local models of the approximating problems. The resulting double-loop structure improves global approximations while adapting local models, providing a flexible and implementable approach for a wide class of composite optimization problems. It also lays the groundwork for new algorithmic developments in this domain.