An interior proximal gradient method for nonconvex optimization

TL;DR

This paper introduces an innovative interior proximal gradient algorithm that effectively handles nonconvex optimization problems with complex constraints, combining interior point and proximal gradient methods to improve convergence and feasibility.

Contribution

The paper proposes a novel algorithm that merges interior point and proximal gradient techniques, providing convergence guarantees for nonconvex problems with inequality constraints.

Findings

Algorithm generates strictly feasible iterates with decreasing objective values.

Convergence results established for fully nonconvex problems.

Refined backtracking technique eliminates the need for stepsize upper bounds.

Abstract

We consider structured minimization problems subject to smooth inequality constraints and present a flexible algorithm that combines interior point (IP) and proximal gradient schemes. While traditional IP methods cannot cope with nonsmooth objective functions and proximal algorithms cannot handle complicated constraints, their combined usage is shown to successfully compensate the respective shortcomings. We provide a theoretical characterization of the algorithm and its asymptotic properties, deriving convergence results for fully nonconvex problems, thus bridging the gap with previous works that successfully addressed the convex case. Our interior proximal gradient algorithm benefits from warm starting, generates strictly feasible iterates with decreasing objective value, and returns after finitely many iterations a primal-dual pair approximately satisfying suitable optimality conditions. As a byproduct of our analysis of proximal gradient iterations we demonstrate that a slight refinement of traditional backtracking techniques waives the need for upper bounding the stepsize sequence, as required in existing results for the nonconvex setting.