Composite Optimization using Local Models and Global Approximations

TL;DR

The paper tackles nonconvex, nonsmooth composite optimization with the form $\min_{x\in X} f_0(x) + h(F(x))$, allowing extended-real-valued $h$ and nonsmooth $F$. It develops a unified double-loop framework that couples global approximations $h^\nu$, $F^\nu$ with locally built convex master problems solved via syntheses of local models, enabling implementable algorithms. A key theoretical result shows that cluster points of near-stationary points of the approximating problems converge to stationary points of the original problem under mild qualification conditions. The approach is demonstrated through motivating examples including buffered failure probability constraints, distributionally robust two-stage programs, and distance-to-set penalties, highlighting the framework's breadth and practical relevance for complex optimization models.

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.