论文标题
统一的布雷格曼交替的最小化算法，用于广义直流编程与成像数据的应用
A Unified Bregman Alternating Minimization Algorithm for Generalized DC Programming with Application to Imaging Data
论文作者
论文摘要
在本文中，我们考虑了一类Nonconvex（不一定是可区分的）优化问题，称为广义DC（通用范围函数）编程，该问题正在最小化两个可分离的DC零件的总和和一个两个块变量耦合函数。为了规避所考虑的问题的非概念性和不可分割性，我们通过最大程度地利用了目标的有利的DC结构来引入统一的Bregman交替最小化算法（UBAMA）。具体而言，我们首先遵循交替最小化的精神，以顺序更新每个块变量，这可以有效地解决由耦合函数引起的非统一性。然后，我们采用Fenchel-Young不平等，近似第二个DC组件（即凹部部分），以便每个子问题都减少到凸优化问题，从而减轻非Concevex DC部分的计算负担。此外，每个子问题都吸收了Bregman近端正规化项，这通常有益于许多情况下通过选择适当的Bregman kernel函数来诱导子问题的封闭式解决方案。值得注意的是，我们的算法不仅提供了一个算法框架，可以理解某些新型现有算法的迭代方案，而且还享受与某些针对通用的非convex和Nonsonsmoothmooth优化问题开发的可实施方案。从理论上讲，我们证明了我们的算法在kurdyka-łojasiewicz（Kł）条件下产生的序列在全球范围内收敛到临界点。此外，当我们进一步了解Kł指数的先前信息时，我们估计算法的局部收敛速率。
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-Łojasiewicz (KŁ) condition. Besides, we estimate the local convergence rates of our algorithm when we further know the prior information of the KŁ exponent.
