Bregman投影梯度法的迭代收敛性研究
On the Iterate Convergence of Bregman Projected Gradient Method
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中文总结 AI 辅助
本文针对带Shannon熵核的Bregman投影梯度法，提出基于缩放Kurdyka-Łojasiewicz性质的收敛分析框架，证明其迭代收敛性及线性收敛条件，为解决该方法迭代收敛的开放问题提供了基础。
中文摘要 AI 辅助
Bregman投影梯度法（BPGM）的迭代收敛性长期以来是一个悬而未决的问题，尤其对于被广泛采用的Shannon熵核而言。现有收敛结果往往存在局限性，依赖于核梯度的Lipschitz连续性或目标函数的限制性条件。本文针对带Shannon熵核的BPGM构建了一种新颖的收敛分析框架，在带线性约束的一大类目标函数下得到了强收敛结果。该框架的核心是一个名为缩放Kurdyka-Łojasiewicz（SKŁ）性质的新概念，它刻画了函数在Bregman几何下的局部增长行为。我们证明SKŁ性质可保证BPGM的迭代收敛，且对所有连续亚解析函数均成立。此外，若问题具有1/2的SKŁ指数，则BPGM序列呈现线性收敛。我们还提供了具有1/2 SKŁ指数的函数示例，证明在严格互补性和目标函数梯度的局部Lipschitz连续性下，1/2的KŁ指数可蕴含1/2的SKŁ指数。基于这些新颖结果，本研究为解决BPGM迭代收敛这一开放问题迈出了第一步。
英文摘要
The iterate convergence of \textit{Bregman projected gradient method} (BPGM) has remained a long-standing open problem, especially for the widely adopted Shannon entropy kernel. Existing convergence results are often limited, relying on Lipschitz continuity of the kernel's gradient or restrictive conditions on the objective function. In this paper, we develop a novel convergence analysis framework for the BPGM with the Shannon entropy kernel, yielding strong convergence results for a broad class of objective functions under linear constraints. The cornerstone of our framework is a new concept called \textit{scaled Kurdyka-Łojasiewicz} (SKŁ) property, which captures the local growth behavior of a function under the Bregman geometry. We show that the SKŁ property ensures the iterate convergence of BPGM and holds for all continuous subanalytical functions. Furthermore, we prove that the BPGM sequence exhibits linear convergence if the problem possesses an SKŁ exponent of $1/2$. We then furnish the examples of functions with the SKŁ exponent $1/2$ by proving that the SKŁ exponent $1/2$ is implied by the KŁ exponent $1/2$ under strict complementarity and local Lipschitz continuity of the objective's gradient. Building on these novel results, our work takes a first step towards resolving the open problem of BPGM iterate convergence.
