New operator designs for Halpern iterations with explicit rates under H\"older error bounds

TL;DR

This paper introduces new operator designs for Halpern iterations that achieve explicit convergence rates under H"older error bounds, improving the efficiency of convex feasibility algorithms.

Contribution

It provides the first explicit convergence rates for Halpern iterations with projection operators under H"older error bounds, extending classical results.

Findings

Convergence rates depend on H"older exponent $\

Halpern iterations outperform Dykstra's algorithm in numerical experiments.

Explicit rates are applicable to common projection operators in convex feasibility problems.

Abstract

We investigate the asymptotic behavior of Halpern-type iterations applied to quasi-nonexpansive operators arising in best approximation problems over the intersection of finitely many closed convex sets in $\mathbb{R}^n$. Assuming a local decrease condition for the underlying operator and standard requirements on the stepsizes $(\alpha_k) \subset (0,1)$, we first prove strong convergence of the Halpern sequence $x_{k+1} = \alpha_k x_0 + (1-\alpha_k) T x_k$ to the best approximation point $x^\star$ in the intersection set, that is, the metric projection of $x_0$ onto that set. Under the additional assumption that the intersection satisfies a H\"older-type error bound with exponent $\gamma \in (0,1]$, we then derive explicit convergence rates for both feasibility and norm error: the distance from $x_k$ to the intersection set decays like $\mathcal O(\alpha_k^{\gamma/(2-\gamma)})$, while the norm error $\|x_k - x^\star\|$ decays like $\mathcal O(\alpha_k^{\gamma/(4-2\gamma)})$. These results apply to most projection-type operators used in convex feasibility problems (including MAP, CRM/SCCRM, Cimmino and 3PM/A3PM) and extend classical convergence analyses of the Halpern-type iterations by providing explicit, geometry-dependent rates governed by H\"older-type error bounds. Our numerical experiments show that Halpern-type iterations combined with most of these projection-type operators are quicker than Dykstra's algorithm to find the projection of a point in an intersection of ellipsoids or in an intersection of polyhedrons.