# First Order Methods beyond Convexity and Lipschitz Gradient Continuity with Applications to Quadratic Inverse Problems

## TL;DR

This paper extends first order methods to nonconvex, nonsmooth problems without requiring Lipschitz gradient continuity, introducing smooth adaptable functions and applying them to quadratic inverse problems with convergence guarantees.

## Contribution

It generalizes the descent lemma and develops a Bregman-based proximal gradient method for nonconvex problems, broadening applicability beyond convex settings.

## Key findings

- Established a full extended descent lemma for nonconvex functions
- Proposed a globally convergent Bregman proximal gradient method
- Applied framework to quadratic inverse problems with new convergence results

## Abstract

We focus on nonconvex and nonsmooth minimization problems with a composite objective, where the differentiable part of the objective is freed from the usual and restrictive global Lipschitz gradient continuity assumption. This longstanding smoothness restriction is pervasive in first order methods (FOM), and was recently circumvent for convex composite optimization by Bauschke, Bolte and Teboulle, through a simple and elegant framework which captures, all at once, the geometry of the function and of the feasible set. Building on this work, we tackle genuine nonconvex problems. We first complement and extend their approach to derive a full extended descent lemma by introducing the notion of smooth adaptable functions. We then consider a Bregman-based proximal gradient methods for the nonconvex composite model with smooth adaptable functions, which is proven to globally converge to a critical point under natural assumptions on the problem's data. To illustrate the power and potential of our general framework and results, we consider a broad class of quadratic inverse problems with sparsity constraints which arises in many fundamental applications, and we apply our approach to derive new globally convergent schemes for this class.