# A Bregman forward-backward linesearch algorithm for nonconvex composite optimization: superlinear convergence to nonisolated local minima

## TL;DR

This paper presents Bella, a Bregman forward-backward splitting algorithm with superlinear convergence for nonconvex composite optimization, leveraging the Bregman forward-backward envelope and linesearch techniques.

## Contribution

Introduction of Bella, a novel superlinearly convergent Bregman splitting method utilizing BFBE and linesearch for nonconvex optimization with nonisolated minima.

## Key findings

- Converges subsequentially to stationary points.
- Achieves superlinear convergence rates under certain conditions.
- Works for nonconvex functions with relative smoothness and nonsmooth components.

## Abstract

We introduce Bella, a locally superlinearly convergent Bregman forward backward splitting method for minimizing the sum of two nonconvex functions, one of which satisfying a relative smoothness condition and the other one possibly nonsmooth. A key tool of our methodology is the Bregman forward-backward envelope (BFBE), an exact and continuous penalty function with favorable first- and second-order properties, and enjoying a nonlinear error bound when the objective function satisfies a Lojasiewicz-type property. The proposed algorithm is of linesearch type over the BFBE along candidate update directions, and converges subsequentially to stationary points, globally under a KL condition, and owing to the given nonlinear error bound can attain superlinear convergence rates even when the limit point is a nonisolated minimum, provided the directions are suitably selected.