Problem
Research questions and friction points this paper is trying to address.
set-valued contractions
computational complexity
fixed points
Nadler's fixed-point theorem
CLS-completeness
On the Complexity of Finding Fixed Points for Set-Valued Contractions
🤖 AI Summary
本文研究了寻找集值压缩映射不动点的计算复杂性，通过构建Projected-Nadler问题并证明其CLS-完全性，进而分析了基本迭代过程的收敛速率。
📝 Abstract
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.
Innovation
Methods, ideas, or system contributions that make the work stand out.
CLS-completeness
Set-valued contractions
Nadler's fixed point theorem
Convergence rate analysis
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