We study the computational complexity of finding fixed points of set-valued contractions. We first formulate a computational problem for Nadler's fixed-point theorem, PROJECTED-NADLER, and prove that it is CLS-complete by showing its equivalence to CONTINUOUS-LOCALOPT. We then establish a stronger converse to Nadler's theorem, which can be applied as a tool to analyze the convergence rate of the set-valued basic iteration procedure. Finally, we reduce the large-margin triplet stationarity problem to PROJECTED-NADLER. Combining the argument for CLS-hardness of exact triplet stationarity problem [Yan et al. 2025] with the CLS-hardness of approximate KKT points of quadratic programs [Fearnley et al. 2025], we obtain CLS-completeness of large-margin triplet stationarity at any prescribed accuracy.

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