Halpern加速的带退化预条件的最优非遍历收敛的majorized ADMM
Halpern-Accelerated Majorized ADMM with Optimal Non-Ergodic Convergence under Degenerate Preconditioning
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中文总结 AI 辅助
本文提出一种Halpern加速的majorized ADMM，允许不定近端项，在退化预条件下实现最优非遍历收敛，并通过数值实验验证其残差衰减与优势。
中文摘要 AI 辅助
我们开发了一种Halpern加速的majorized交替方向乘子法（ADMM），该方法可能带有不定近端项，用于线性约束凸复合优化。Majorization简化了子问题，但引入了对标准退化近端点表示的前向扰动。我们将majorized ADMM重新表述为一种扰动的退化近端点方法，并将诱导映射的不动点刻画为Karush--Kuhn--Tucker（KKT）解。在范围兼容性和度量余强单调性条件下，该映射在由可能奇异的预条件子诱导的半范数下是非扩张的。在有限维中，一个可验证的块矩阵条件允许不定近端正则化，同时保持所需的度量性质。松弛的Halpern迭代随后产生一个$\mathcal O(1/k)$的不动点残差界，在一般非扩张设置中是最优的，以及在中间迭代处产生一个$\mathcal O(1/k)$的非遍历KKT残差界。数值实验展示了预测的残差衰减以及majorization和不定近端项的好处。
英文摘要
We develop a Halpern-accelerated majorized alternating direction method of multipliers (ADMM) with possibly indefinite proximal terms for linearly constrained convex composite optimization. Majorization simplifies the subproblems but introduces a forward perturbation to the standard degenerate proximal point representation. We reformulate majorized ADMM as a perturbed degenerate proximal point method and characterize the fixed points of the induced mapping as Karush--Kuhn--Tucker (KKT) solutions. Under range compatibility and metric cocoercivity conditions, the mapping is nonexpansive in the seminorm induced by a possibly singular preconditioner. In finite dimensions, a verifiable block-matrix condition permits indefinite proximal regularization while retaining the required metric properties. A relaxed Halpern iteration then yields an $\mathcal O(1/k)$ fixed-point residual bound, sharp in the general nonexpansive setting, and an $\mathcal O(1/k)$ nonergodic KKT residual bound at the intermediate iterates. Numerical experiments illustrate the predicted residual decay and the benefits of majorization and indefinite proximal terms.
发表机构
- Dongguan University of Technology（东莞理工学院）
- Peking University（北京大学）
- Northeastern University（东北大学）
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