Composite Optimization using Local Models and Global Approximations

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.