Variable Bregman Majorization-Minimization algorithms for nonconvex nonsmooth optimization, with application to Poisson imaging

TL;DR

This paper introduces a flexible Bregman majorization-minimization framework for nonconvex nonsmooth optimization, with convergence guarantees and applications to Poisson imaging, demonstrated through PET image reconstruction.

Contribution

It develops a unifying variable Bregman MM algorithm with convergence analysis under Kurdyka-Lojasiewicz property, extending existing methods to non-Lipschitz scenarios.

Findings

Converges to critical points under broad conditions.

Constructs tractable surrogate functions for complex problems.

Successfully applied to PET image reconstruction with nonconvex regularizers.

Abstract

In this work, we introduce a unifying Bregman-based majorization-minimization (MM) framework for solving nonconvex nonsmooth optimization problems. The proposed approach leverages Bregman divergences, possibly varying across iterations, to construct tailored surrogate functions that majorize the objective. We establish the convergence of the iterates of the resulting variable Bregman MM algorithm to critical points under the Kurdyka-Lojasiewicz property, relaxing standard assumptions such as the Lipschitz smoothness of the nonconvex objective function. We derive a constructive methodology to build a broad class of variable Bregman majorants with tractable minimizers. Our study encapsulates various existing majorization techniques, in particular those derived for Poisson data fidelity terms in imaging inverse problems. Numerical experiments on Positron Emission Tomography (PET) image reconstruction with a nonconvex regularizer showcase the practical feasibility of the proposed scheme.