Whole-Sequence Convergence of Variable-Smoothing Full-Splitting Methods via a Lifted Kurdyka-Łojasiewicz Framework
Abstract
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 γk=(k+k0)−β, with k0≥1 and 1/2<β≤1, 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.
Related
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BibTeX
@misc{tao2026wholesequence,
title = {{Whole-Sequence Convergence of Variable-Smoothing Full-Splitting Methods via a Lifted Kurdyka-Łojasiewicz Framework}},
author = {Min Tao},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.28306},
eprint = {2609.28306},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.28306},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Whole-Sequence Convergence of Variable-Smoothing Full-Splitting Methods via a Lifted Kurdyka-Łojasiewicz Framework
AU - Min Tao
PY - 2026
DA - 2026/09/23
PB - arXiv
DO - 10.48550/arxiv.2609.28306
UR - https://arxiv.org/abs/2609.28306
AB - 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≥1 and 1/2<β≤1, 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.
ER -