New operator designs for Halpern iterations with explicit rates under Hölder error bounds
Pablo Barros, Vincent Guigues, Roger Behling, Luiz-Rafael Santos
TL;DR
This paper analyzes Halpern-type anchored iterations $x_{k+1}=\alpha_k x_0+(1-\alpha_k) T x_k$ for the Best Approximation Problem over intersections of convex sets, proving strong convergence to the metric projection $P_S(x_0)$ under a local decrease condition on the operator $T$ and standard stepsize rules. It establishes explicit convergence rates governed by a Hölder error bound with exponent $\gamma\in(0,1]$, yielding $\mathfrak d(x_k)=\mathcal O(\alpha_k^{\gamma/(2-\gamma)})$ for feasibility and $\|x_k-P_S(x_0)\|=\mathcal O(\alpha_k^{\gamma/(4-2\gamma)})$ for the norm error. The authors verify that six projection-based operators (MAP, Cimmino, 3PM, A3PM, SCCRM, and CRM in product space) satisfy the key Assumption [ass:T], enabling the same global convergence and rate results across these methods, including cases where the operators are not globally nonexpansive. Numerical experiments show that Halpern-type iterations often outperform Dykstra's algorithm in both ellipsoidal and polyhedral intersections, highlighting the practical impact of the derived rates and the versatility of the approach.
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.