AI 中文总结
研究结构化凸优化问题，改进自适应扩展黄金比例原始对偶算法，证明原始步长上限冗余，无需步长上限、线搜索等，建立线性收敛，开发加速变体并证明遍历\(\mathcal O(1/N^2)\)收敛速度，数值实验验证方法有效性。
AI 中文摘要
本文重新审视了Soe等人（2026年）提出的用于涉及仅局部光滑的可微项的结构化凸优化问题的自适应扩展黄金比例原始对偶算法（aEGRPDA）。我们证明了aEGRPDA中对原始步长施加的人工上限是多余的，因为自适应规则本身使步长有上界。结果，目标残差和可行性违反的遍历\(\mathcal O(1/N)\)估计与该超参数无关。因此，所得的自适应黄金比例原始对偶方法既不需要步长上限，也不需要线搜索过程，也不需要已知的全局Lipschitz常数。当原始函数和对偶函数都是强凸函数时，我们建立了算法的线性收敛性。此外，除了局部光滑性假设外，我们还开发了两种加速变体：一种用于非光滑原始分量是强凸的情况，另一种用于可微项是全局强凸的情况。对于这些加速方法，我们证明了遍历\(\mathcal O(1/N^2)\)收敛速度。在泊松成像问题上的初步数值实验说明了所提出方法的效率和鲁棒性。
英文摘要
This paper revisits the adaptive extended golden-ratio primal--dual algorithm (aEGRPDA) proposed by Soe et al. (2026) for structured convex optimisation problems involving a differentiable term that is only locally smooth. We prove that the artificial upper bound imposed on the primal step-size in aEGRPDA is redundant, since the adaptive rule itself keeps the step-sizes bounded above. As a consequence, the ergodic $\mathcal O(1/N)$ estimates for the objective residual and feasibility violation, where $N\ge1$ denotes the number of iterations, are independent of this hyperparameter. Consequently, the resulting adaptive golden-ratio primal--dual method, therefore, requires neither a step-size cap, nor a linesearch procedure, nor a known global Lipschitz constant. We establish linear convergence of the algorithm when both the primal and dual functions are strongly convex. Furthermore, we develop two accelerated variants, in addition to the local smoothness assumption: one for the case where the nonsmooth primal component is strongly convex, and another for the case where the differentiable term is globally strongly convex. For these accelerated methods, we prove an ergodic $\mathcal O(1/N^2)$ convergence rate. Preliminary numerical experiments on a Poisson imaging problem illustrate the efficiency and robustness of the proposed approaches.
Comments33 pages
