Abstract
In the setting of operatorname{CAT}(kappa) spaces, common fixed point iterations built from prox mappings (e.g. prox-prox, Krasnoselsky–Mann relaxations, nonlinear projected-gradients) converge locally linearly under the assumption of linear metric subregularity. Linear metric subregularity is in any case necessary for linearly convergent fixed point sequences, so the result is tight. To show this, we develop a theory of fixed point mappings that violate the usual assumptions of nonexpansiveness and firm nonexpansiveness in p-uniformly convex spaces.
Highlights
Examples of p-uniformly convex spaces are Lp spaces, CAT(0) spaces (p = c = 2), Hadamard spaces (complete CAT(0) spaces), Hilbert spaces
1 Fundamentals of nonlinear spaces Following [1] we focus on p-uniformly convex spaces with parameter c [12]: for p ∈ (1, ∞), a metric space (G, d) is p-uniformly convex with constant c > 0 whenever it is a geodesic space, and
If T is pointwise almost α-firmly nonexpansive at y ∈ Fix T with arbitrarily small violation, whenever T is metrically subregular at y, there is a neighborhood of y on which convergence of the fixed point iteration can be quantified by the said gauge
Summary
Lemma 17 (Gauge monotonicity and almost quasi α-firmness implies convergence to fixed points) Let (G, d) be a complete p-uniformly convex space with constant c. If T is pointwise almost α-firmly nonexpansive at y ∈ Fix T with arbitrarily small violation , whenever T is (gauge) metrically subregular at y, there is a neighborhood of y on which convergence of the fixed point iteration can be quantified by the said gauge.
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