Abstract:
We propose a unifying algorithm for non-smooth non-convex optimization. The algorithm approximates the objective function by a convex model function and finds an approximate (Bregman) proximal point of the convex model. This approximate minimizer of the model function yields a descent direction, along which the next iterate is found. Complemented with an Armijo-like line search strategy, we obtain a flexible algorithm for which we prove (subsequential) convergence to a stationary point under weak assumptions on the growth of the model function error.
Special instances of the algorithm with a Euclidean distance function are, for example, Gradient Descent, Forward--Backward Splitting, ProxDescent, without the common requirement of a "Lipschitz continuous gradient". In addition, we consider a broad class of Bregman distance functions (generated by Legendre functions) replacing the Euclidean distance. The algorithm has a wide range of applications including many linear and non-linear inverse problems in image processing and machine learning.

Bibtex: @techreport{OFB17,
title = {Non-smooth Non-convex Bregman Minimization: Unification and new Algorithms},
author = {P. Ochs and J. Fadili and T. Brox},
year = {2017},
journal = {ArXiv e-prints, arXiv:1707.02278 [math.OC]},
}