发表机构
Sun Yat-sen University; National University of Singapore; University of Tennessee, Knoxville（中山大学; 新加坡国立大学; 田纳西大学诺克斯维尔分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出一种非精确 Halpern 加速预条件广义近端点算法，结合锚定、非精确预解与全范围松弛，求解极大单调包含问题，并给出 O(1/k²) 收敛率及 pADMM 和 PDHG 的非精确加速版本。
AI 中文摘要
本文研究了一种带有可容许半正定预条件子 $\mathcal{M}$ 的非精确 Halpern 加速广义近端点算法，用于求解极大单调包含问题。所提出的框架结合了 Halpern 锚定、预解非精确性以及全权重范围 $\rho\in(0,2]$ 上的松弛，从而在统一方案中涵盖了欠松弛和过松弛近端迭代。我们首先在允许一般锚定参数和某些不可和容差的条件下，建立了非精确预解序列的收敛性。对于锚定参数 $\beta_{k}=1/(k+r)$ 且 $r\geq2$，我们随后推导了 $\mathcal{M}$-半范数下平方不动点残差的显式界。特别地，若容差满足 $\varepsilon_{k}=\mathcal{O}((k+1)^{-\alpha})$，其中当 $0<\rho<2$ 时 $\alpha>3/2$，当 $\rho=2$ 时 $\alpha>2$，这些界产生 $\mathcal{O}(1/k^{2})$ 的收敛速率。在 $\rho=2$ 处对非精确容差更强的衰减条件，突出了非精确设置中端点与内部区域之间的定性区别。最后，我们基于该框架开发了预条件交替方向乘子法（pADMM）和预条件原始-对偶混合梯度（PDHG）方法的非精确加速版本，推导了基于子问题残差的非精确性准则，并为其非精确迭代建立了非遍历的 $\mathcal{O}(1/k)$ KKT 残差速率。
英文摘要
This paper studies an inexact Halpern-accelerated generalized proximal point algorithm with an admissible positive semidefinite preconditioner $\mathcal{M}$ for solving maximal monotone inclusion problems. The proposed framework combines Halpern anchoring, resolvent inexactness, and relaxation over the full weight range $ρ\in(0,2]$, thereby covering both under-relaxed and over-relaxed proximal iterations within a unified scheme. We first establish convergence of the inexact resolvent sequence under conditions that allow general anchoring parameters and certain nonsummable tolerances. For anchoring parameters $β_{k}=1/(k+r)$ with $r\geq2$, we then derive explicit bounds on the squared fixed-point residual in the $\mathcal{M}$-seminorm. In particular, if the tolerances satisfy $\varepsilon_{k}=\mathcal{O}((k+1)^{-α})$ with $α>3/2$ for $0<ρ<2$ and $α>2$ for $ρ=2$, these bounds yield an $\mathcal{O}(1/k^{2})$ convergence rate. The stronger decay condition on the inexactness tolerances at $ρ=2$ highlights a qualitative distinction between the endpoint and the interior regime in the inexact setting. Finally, we develop inexact accelerated versions of the preconditioned alternating direction method of multipliers (pADMM) and the preconditioned primal--dual hybrid gradient (PDHG) method based on this framework, derive inexactness criteria based on subproblem residuals, and establish a nonergodic $\mathcal{O}(1/k)$ KKT residual rate for their inexact iterates.
