变光滑全分裂方法中全序列收敛性的失效
Counterexamples to Whole-Sequence Convergence of Variable-Smoothing Full-Splitting Methods
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中文总结 AI 辅助
本文构造实例证明，用于非凸非光滑优化的变光滑全分裂方法S-FSPS不满足全序列收敛性，仅存在子序列收敛，为该类方法的收敛性研究提供了反例。
中文摘要 AI 辅助
我们研究由Boţ、Li和Tao（发表于《SIAM J. Optim.》，2025年，第35卷第4期，第2623-2653页）提出的、用于结构化非凸非光滑分式规划的基于光滑化的全分裂邻近次梯度方法（S-FSPS，即算法4.1）的全序列收敛性。现有收敛理论仅证明了存在子序列收敛到精确极限提升平稳点，而整个生成序列是否必须收敛仍未明确。我们通过构造两个可容许实例否定了该问题：其原始序列的聚点集为{1}×S¹，轨迹长度无限，尽管每个聚点均为精确极限提升平稳点。第一个构造中，线性算子A非零，且每个生成的对偶迭代均非零；第二个构造中，可行集满维，A满行秩，且f∘K与g∘A在可行集上均非恒定。第一个构造还为对应的变光滑、单循环、全分裂非凸非光滑复合优化方法提供了一个不收敛的示例，尽管其子序列收敛到精确极限平稳点。据我们所知，这是首个明确构造表明，对于变光滑全分裂方案，即使存在消失的非可和光滑化，且结合精确极限平稳性，也不蕴含全序列收敛性。
英文摘要
We study whole-sequence convergence of the smoothing-based full-splitting proximal subgradient method (S-FSPS) for structured nonconvex and nonsmooth fractional programs, introduced by Boţ, Li, and Tao (SIAM J. Optim., 35(4):2623--2653, 2025) as Algorithm~4.1. Existing theory guarantees only the existence of a subsequence converging to a limiting lifted stationary point. We show that this guarantee is sharp by constructing two admissible instances whose corresponding primal sequences both have cluster set $\{1\}\times\mathbb S^1$ and infinite length, although every cluster point is a limiting lifted stationary point. The construction prescribes a slowly rotating spiral and realizes it exactly through a compatible first-order jet and a $C^{1,1}$ Whitney extension. In the first instance, $A$ has rank one and every smoothing-dual iterate is nonzero. In the second, the feasible set is full-dimensional, $A$ has full row rank, and each of $f\circ K$, $g\circ A$, and the numerator $g\circ A+h$ is nonconstant on the feasible set. The first instance also yields a nonconvergent example for the corresponding variable-smoothing, single-loop, full-splitting method for nonconvex and nonsmooth composite optimization, although that method still admits a subsequence converging to an exact stationary point. Thus, a vanishing but nonsummable smoothing schedule does not imply whole-sequence convergence.
