New operator designs for Halpern iterations with explicit rates under Hölder error bounds

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 $(α_k) \subset (0,1)$, we first prove strong convergence of the Halpern sequence $x_{k+1} = α_k x_0 + (1-α_k) T x_k$ 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 Hölder-type error bound with exponent $γ\in (0,1]$, we then derive explicit convergence rates for both feasibility and norm error: the distance from $x_k$ to the intersection set decays like $\mathcal O(α_k^{γ/(2-γ)})$, while the norm error $\|x_k - x^\star\|$ decays like $\mathcal O(α_k^{γ/(4-2γ)})$. 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 Hölder-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.