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

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The study establishes explicit convergence rates for Halpern-type iterations applied to quasi-nonexpansive operators under H\"older error bounds, showing that the distance from $x_k$ to the intersection set and the norm error $\|x_k - x^\star\|$ decay at specific rates depending on the stepsize $(\alpha_k)$ and the exponent $\gamma$. This analysis provides faster convergence than Dykstra's algorithm for projecting points onto intersections of ellipsoids or polyhedrons using various projection-type operators.

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arXiv:2601.14451v3 Announce Type: replace 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.