关于集值压缩映射不动点寻找的复杂性
On the Complexity of Finding Fixed Points for Set-Valued Contractions
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中文总结 AI 辅助
本文通过证明Projected-Nadler问题的CLS完全性，并归约大间隔三元组平稳性问题，揭示了集值压缩映射不动点寻找的计算复杂性。
中文摘要 AI 辅助
本文研究了寻找集值压缩映射不动点的计算复杂性。我们首先为Nadler不动点定理制定了一个计算问题：Projected-Nadler，并通过证明其与Continuous-LocalOpt的等价性，证明了该问题是$\mathsf{CLS}$-完全的。接着，我们为Nadler不动点定理建立了一个更强的逆命题，该命题可作为分析集值基本迭代过程收敛速度的工具。最后，我们将大间隔三元组平稳性问题归约到Projected-Nadler。结合[arXiv:2509.16898]中引入的$\mathsf{CLS}$-困难性，这得到了大间隔三元组平稳性的$\mathsf{CLS}$-完全性。
英文摘要
In this paper, we study the computational complexity of finding fixed points for set-valued contractions. We first formulate a computational problem for Nadler's fixed-point theorem: Projected-Nadler, and prove that it is $\mathsf{CLS}$-complete by showing its equivalence to Continuous-LocalOpt. We then establish a stronger converse for Nadler's fixed point theorem that can be applied as a tool to analyze the convergence rate of set-valued basic iteration procedure. Finally, we reduce large-margin triplet stationarity problem to Projected-Nadler. Together with its $\mathsf{CLS}$-hardness introduced in [arXiv:2509.16898], this yields $\mathsf{CLS}$-completeness of large-margin triplet stationarity.
发表机构
- University of Wisconsin-Madison（威斯康星大学麦迪逊分校）
机构由 AI 辅助整理，请以论文原文为准。
