Polynomial-time computation of $\ell_p$-contraction fixed points for even $p$

2026-07-26

AI Summary

This work proposes the first algorithm capable of computing an $\varepsilon$-approximate fixed point of an $\ell_p$-nonexpansive mapping in time polynomial in $d$ and $\log(1/\varepsilon)$, for any fixed even integer $p \neq 2$. The approach leverages an efficiently computable version of Sion’s minimax theorem tailored to non-compact settings, integrating techniques from convex optimization with the analysis of low-degree polynomial potential functions. This framework extends fixed-point computation to a broader class of potential-driven search problems. By achieving logarithmic dependence on the approximation accuracy $\varepsilon$, the method significantly improves the efficiency of fixed-point computation in high-dimensional non-Euclidean spaces, establishing the first polynomial-time algorithm with such precision dependence.

Abstract

We give a $\text{poly}(d, p, \log(1/ε))$-time algorithm that computes an $ε$-approximate fixed point of any $\ell_p$-nonexpansive map $f : \mathcal{X} \to \mathcal{X}$, where $\mathcal{X} \subset \mathbb R^d$ is a convex compact set and $p$ is an even integer. This is the first algorithm with $\text{poly}(d, \log(1/ε))$ runtime for any fixed $p \ne 2$. Our techniques are based on a computationally efficient version of Sion's theorem for non-compact minmax problems, and extend to more general total search problems that admit low-degree polynomial potentials.

Problem

• fixed point
• ℓ_p-norm
• nonexpansive map
• polynomial-time computation
• even p

Innovation

• fixed point
• ℓ_p-nonexpansive map
• polynomial-time algorithm
• Sion's theorem
• minmax problems