← 研踪 PaperTrailarXiv:2609.14101 · v3
On the Complexity of Finding Fixed Points for Set-Valued Contractions
首次提交 2026-09-13 02:53
版本更新 2026-10-29 12:21
雷达观测 2026-10-30 11:25
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.
作者备注：58 pages, 4 figures
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v3 · 2026-10-29 12:21 · 最新观测版本
On the Complexity of Finding Fixed Points for Set-Valued Contractions
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.
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