A Nonlinear Bregman Primal-Dual Framework for Optimizing Nonconvex Infimal Convolutions

TL;DR

This paper introduces a novel nonlinear Bregman primal-dual framework for optimizing complex nonconvex infimal convolution problems, unifying and extending existing algorithms with proven convergence and promising experimental results.

Contribution

It proposes a new block coordinate descent scheme on a Bregman augmented Lagrangian for nonconvex inf-conv problems, including convergence guarantees and broad applicability.

Findings

Algorithm converges to stationary points.

Performs favorably compared to DC-programming and k-means.

Exhibits robustness to initialization.

Abstract

This work is concerned with the optimization of nonconvex, nonsmooth composite optimization problems, whose objective is a composition of a nonlinear mapping and a nonsmooth nonconvex function, that can be written as an infimal convolution (inf-conv). To tackle this problem class we propose to reformulate the problem exploiting its inf-conv structure and derive a block coordinate descent scheme on a Bregman augmented Lagrangian, that can be implemented asynchronously. We prove convergence of our scheme to stationary points of the original model for a specific choice of the penalty parameter. Our framework includes various existing algorithms as special cases, such as DC-programming, $k$-means clustering and ADMM with nonlinear operator, when a specific objective function and inf-conv decomposition (inf-deconvolution) is chosen. In illustrative experiments, we provide evidence, that our algorithm behaves favorably in terms of low objective value and robustness towards initialization, when compared to DC-programming or the $k$-means algorithm.