求解一类复合优化的惯性近似Bregman近端梯度算法
Inertial Approximate Bregman Proximal Gradient Algorithms for a Class of Composite Optimization Problems
摘要: 本文提出了一种惯性近似Bregman近端梯度算法针对目标函数梯度不满足全局Lipschitz连续的复合优化问题。通过利用相对光滑来代替全局Lipschitz条件，引入惯性外推机制以提升收敛速度，并结合一种回溯线搜索，确保非凸环境下的数值稳定性。此外，我们采用了一种近似Bregman距离的新度量，使得子问题的求解比现有Bregman类算法更为简单。在收敛性证明中，在目标函数满足Kurdyka-Łojasiewicz (KL)性质的假设下，证明了算法生成的序列具有有限长度，并全局收敛至广义平稳点。
Abstract: This paper proposes an inertial approximate Bregman proximal gradient algorithm for composite optimization problems where the gradient of the objective function does not satisfy global Lipschitz continuity. By utilizing relative smoothness to replace the global Lipschitz condition, an inertial extrapolation mechanism is introduced to improve the convergence rate, and combined with a backline search, numerical stability in non-convex settings is ensured. Furthermore, we adopt a new metric for the approximate Bregman distance, which simplifies the solution of the subproblems compared to existing Bregman-type algorithms. In the convergence proof, under the assumption that the objective function satisfies the Kurdyka-Łojasiewicz (KL) property, we prove that the sequence generated by the algorithm is of finite length and converges globally to a generalized steady point.
文章引用：房宛玉. 求解一类复合优化的惯性近似Bregman近端梯度算法[J]. 应用数学进展, 2026, 15(5): 418-434. https://doi.org/10.12677/aam.2026.155240

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