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arXiv · 2609.14101

On the Complexity of Finding Fixed Points for Set-Valued Contractions

Emmanouil-Vasileios Vlatakis-Gkaragkounis·Pucheng Xiong
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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.

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Emmanouil-Vasileios Vlatakis-Gkaragkounis, Pucheng Xiong. 2026-09-12. On the Complexity of Finding Fixed Points for Set-Valued Contractions. https://arxiv.org/abs/2609.14101

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