Computational Problems in Metric Fixed Point Theory and their Weihrauch Degrees

TL;DR

This paper investigates the computational complexity of fixed point problems in Banach and Hilbert spaces, revealing their equivalence to well-known computational principles and the limitations of computability in infinite dimensions.

Contribution

It characterizes the Weihrauch degrees of fixed point theorems in uniformly convex Banach spaces and explores the computability of fixed points and convergence rates in infinite-dimensional spaces.

Findings

Fixed point sets correspond to nonempty, co-r.e. weakly closed convex sets.

The Weihrauch degree of the Browder-Goehde-Kirk theorem matches a closed choice principle.

Computable fixed points exist in finite dimensions but not necessarily in infinite dimensions.

Abstract

We study the computational difficulty of the problem of finding fixed points of nonexpansive mappings in uniformly convex Banach spaces. We show that the fixed point sets of computable nonexpansive self-maps of a nonempty, computably weakly closed, convex and bounded subset of a computable real Hilbert space are precisely the nonempty, co-r.e. weakly closed, convex subsets of the domain. A uniform version of this result allows us to determine the Weihrauch degree of the Browder-Goehde-Kirk theorem in computable real Hilbert space: it is equivalent to a closed choice principle, which receives as input a closed, convex and bounded set via negative information in the weak topology and outputs a point in the set, represented in the strong topology. While in finite dimensional uniformly convex Banach spaces, computable nonexpansive mappings always have computable fixed points, on the unit ball in infinite-dimensional separable Hilbert space the Browder-Goehde-Kirk theorem becomes Weihrauch-equivalent to the limit operator, and on the Hilbert cube it is equivalent to Weak Koenig's Lemma. In particular, computable nonexpansive mappings may not have any computable fixed points in infinite dimension. We also study the computational difficulty of the problem of finding rates of convergence for a large class of fixed point iterations, which generalise both Halpern- and Mann-iterations, and prove that the problem of finding rates of convergence already on the unit interval is equivalent to the limit operator.