The value function approach to convergence analysis in composite optimization

TL;DR

This paper analyzes the convergence of first-order local search methods for complex composite optimization problems, introducing a general framework based on the value function and tameness assumptions, applicable to many practical problems.

Contribution

It provides a novel convergence analysis of the composite Gauss-Newton method under tameness conditions, extending the understanding of sequential convex programming.

Findings

Convergence analysis applicable to a broad class of problems

Extension of semi-algebraic assumptions to tameness

Applicable to practical problems with complex geometries

Abstract

This works aims at understanding further convergence properties of first order local search methods with complex geometries. We focus on the composite optimization model which unifies within a simple formalism many problems of this type. We provide a general convergence analysis of the composite Gauss-Newton method under tameness assumptions (an extension of semi-algebraicity). Tameness is a very general condition satisfied by virtually all problems solved in practice. The analysis is based on recent progresses in understanding convergence properties of sequential convex programming methods through the value function.