Mathematical Logic Seminar

# The strange case of Dykstra’s algorithm

Videoconferência

Por Pedro Pinto (Technische Universität Darmstadt).

In this talk we discuss a proof mining treatment of the strong convergence of Dykstra’s algorithm.
Halpern’s iterative method is probably the most common approach to strongly approximate fixed points of nonexpansive maps. The canonical proof (pliable to many other results) establishing strong convergence of the iteration relies on a crucial use of sequential weak compactness. It is well-understood how a proof-theoretical approach allows for the elimination of the compactness arguments via bounded collection principles, thus allowing for simple quantitative data in the analysis of such proofs.
Here, we focus on a different iterative method. Generalizing the alternating projection method, Dykstra’s algorithm strongly approximates the optimal solution of the convex feasibility problem. Similarly to Halpern, the strong convergence of Dykstra’s method makes crucial uses of compactness principles substantiated by arithmetical comprehension. Yet, as the iterative schema has no connection with Halpern’s definition and the proof follows a completely different structure, it was not known whether the removal of the compactness arguments would be possible and thus, a priori, we were only guaranteed to obtain quantitative data defined by bar-recursive functionals. Strikingly, still here, it was possible to bypass the use of arithmetical comprehension and bar-recursive functionals. We will discuss the recent quantitative analysis of Dykstra’s convergence proof and explain how it was possible to avoid the compactness principles crucial in the original proof.

References:

• [1] F. Ferreira, L. Leustean, P. Pinto, On the removal of weak compactness arguments in proof mining, Advances in Mathematics 354, 55pp, 2019.
• [2] U. Kohlenbach, P. Pinto, Fejér monotone sequences revisited, 2023 (preprint available at homepage https://www2.mathematik.tu-darmstadt.de/~pinto/).
• [3] P. Pinto, On the finitary content of Dykstra’s cyclic projections algorithm, 2023 (preprint available at homepage).

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