发表机构
School of Mathematics and National Key Laboratory for Novel Software Technology, Nanjing University（南京大学数学学院与新型软件技术国家重点实验室）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文通过构造固定提升势函数和缩放KL论证，证明了变平滑全分裂方法在非可和参数下的全序列收敛性，无需全行秩假设，并推广至投影梯度方法。
AI 中文摘要
我们建立了Boţ、Li和Tao（SIAM J. Optim., 35 (2025), pp. 2623–2653）提出的基于平滑的全分裂近端次梯度方法（S-FSPS）在给定非可和、递减的平滑参数序列下原始迭代的全序列收敛性。挑战在于每次迭代使用不同的平滑模型。我们通过构造固定的提升势函数，推导兼容的充分下降和相对误差估计，并应用一个缩放的Kurdyka–Łojasiewicz（KL）有限长度论证来控制由平滑参数变化引起的残差来解决这一问题。在对应的固定提升势函数的KL假设以及这些残差的加权可和性条件下，原始轨迹具有有限长度。对于幂次调度γ_k=(k+k_0)^{-β}，其中k_0≥1且1/2<β≤1，当提升指数足够大时，可和性条件成立。因此，原始序列收敛到一个精确的极限提升平稳点，而无需对线性算子施加全行秩假设。作为推论，我们在非凸非光滑复合优化中建立了变平滑全分裂投影梯度方法的全序列收敛性。例子区分了原始收敛与辅助对偶变量的收敛，并说明了非可和性在保证精确平稳性中的作用。
英文摘要
We establish whole-sequence convergence of the primal iterates of the smoothing-based full-splitting proximal subgradient method (S-FSPS) of Boţ, Li, and Tao (SIAM J. Optim., 35 (2025), pp.~2623--2653) with a prescribed, nonsummable sequence of vanishing smoothing parameters. The challenge is that each iteration uses a different smoothed model. We address this by constructing fixed lifted potentials, deriving compatible sufficient-decrease and relative-error estimates, and applying a scaled Kurdyka--Łojasiewicz (KL) finite-length argument that controls the residuals caused by changes in the smoothing parameter. Under a KL assumption on the corresponding fixed lifted potential and a weighted summability condition on these residuals, the primal trajectory has finite length. For power schedules $γ_k=(k+k_0)^{-β}$, with $k_0\geq1$ and $1/2<β\leq1$, the summability condition holds when the lift exponent is sufficiently large. Consequently, the primal sequence converges to an exact limiting lifted stationary point without imposing full-row-rank assumptions on the linear operators. As a corollary, we establish whole-sequence convergence for a variable-smoothing full-splitting projected-gradient method in nonconvex nonsmooth composite optimization. Examples distinguish primal convergence from convergence of the auxiliary dual variables and illustrate the role of nonsummability in guaranteeing exact stationarity.
