A Unified Bregman Alternating Minimization Algorithm for Generalized DC Programming with Application to Imaging Data

TL;DR

This paper introduces a unified algorithm for solving complex nonconvex optimization problems involving difference-of-convex functions, with applications to imaging data, ensuring convergence to critical points and simplifying subproblem solutions.

Contribution

The paper proposes a novel Unified Bregman Alternating Minimization Algorithm (UBAMA) that efficiently handles generalized DC programming problems by exploiting their structure and ensuring convergence.

Findings

Algorithm guarantees global convergence to critical points.

Subproblems are easier to solve due to Bregman regularization.

Applicable to imaging data with improved computational efficiency.

Abstract

In this paper, we consider a class of nonconvex (not necessarily differentiable) optimization problems called generalized DC (Difference-of-Convex functions) programming, which is minimizing the sum of two separable DC parts and one two-block-variable coupling function. To circumvent the nonconvexity and nonseparability of the problem under consideration, we accordingly introduce a Unified Bregman Alternating Minimization Algorithm (UBAMA) by maximally exploiting the favorable DC structure of the objective. Specifically, we first follow the spirit of alternating minimization to update each block variable in a sequential order, which can efficiently tackle the nonseparablitity caused by the coupling function. Then, we employ the Fenchel-Young inequality to approximate the second DC components (i.e., concave parts) so that each subproblem reduces to a convex optimization problem, thereby alleviating the computational burden of the nonconvex DC parts. Moreover, each subproblem absorbs a Bregman proximal regularization term, which is usually beneficial for inducing closed-form solutions of subproblems for many cases via choosing appropriate Bregman kernel functions. It is remarkable that our algorithm not only provides an algorithmic framework to understand the iterative schemes of some novel existing algorithms, but also enjoys implementable schemes with easier subproblems than some state-of-the-art first-order algorithms developed for generic nonconvex and nonsmooth optimization problems. Theoretically, we prove that the sequence generated by our algorithm globally converges to a critical point under the Kurdyka-{\L}ojasiewicz (K{\L}) condition. Besides, we estimate the local convergence rates of our algorithm when we further know the prior information of the K{\L} exponent.