Browsing by Author "Pinto, Pedro"

Now showing 1 - 3 of 3
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    On Dykstra's Algorithm with Bregman Projections
    (Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2024) Pinto, Pedro; Pischke, Nicholas
    We provide quantitative results on the asymptotic behavior of Dykstra's algorithm with Bregman projections, a combination of the well-known Dykstra's algorithm and the method of cyclic Bregman projections, designed to find best approximations and solve the convex feasibility problem in a non-Hilbertian setting. The result we provide arise through the lens of proof mining, a program in mathematical logic which extracts computational information from non-effective proofs. Concretely, we provide a highly uniform and computable rate of metastability of low complexity and, moreover, we also specify general circumstances in which one can obtain full and effective rates of convergence. As a byproduct of our quantitative analysis, we also for the first time establish the strong convergence of Dykstra's method with Bregman projections in infinite dimensional (reflexive) Banach spaces.
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    Proof Mining and the Convex Feasibility Problem : the Curious Case of Dykstra's Algorithm
    (Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2024) Pinto, Pedro
    In a recent proof mining application, the proof-theoretical analysis of Dykstra's cyclic projections algorithm resulted in quantitative information expressed via primitive recursive functionals in the sense of Gödel. This was surprising as the proof relies on several compactness principles and its quantitative analysis would require the functional interpretation of arithmetical comprehension. Therefore, a priori one would expect the need of Spector's bar-recursive functionals. In this paper, we explain how the use of bounded collection principles allows for a modified intermediate proof justifying the finitary results obtained, and discuss the approach in the context of previous eliminations of weak compactness arguments in proof mining.
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    The Alternating Halpern-Mann Iteration for Families of Maps
    (Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2024) Firmino, Paulo; Pinto, Pedro
    We generalize the alternating Halpern-Mann iteration to countably infinite families of nonexpansive maps and prove its strong convergence towards a common fixed point in the general nonlinear setting of Hadamard spaces. Our approach is based on a quantitative perspective which allowed to circumvent prevalent troublesome arguments and in the end provide a simple convergence proof. In that sense, discussing both the asymptotic regularity and the strong convergence of the iteration in quantitative terms, we furthermore provide low complexity uniform rates of convergence and of metastability (in the sense of T. Tao). In CAT(0) spaces, we obtain linear and quadratic uniform rates of convergence. Our results are made possible by proof-theoretical insights of the research program proof mining and extend several previous theorems in the literature.