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Portugaliae Mathematica

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Volume 77, Issue 3/4, 2020, pp. 345–381
DOI: 10.4171/PM/2054

Published online: 2020-12-22

Metastability of the proximal point algorithm with multi-parameters

Bruno Dinis[1] and Pedro Pinto[2]

(1) Universidade de Lisboa, Portugal
(2) Technische Universität Darmstadt, Germany

In this article we use techniques of proof mining to analyse a result, due to Yonghong Yao and Muhammad Aslam Noor, concerning the strong convergence of a generalized proximal point algorithm which involves multiple parameters. Yao and Noor’s result ensures the strong convergence of the algorithm to the nearest projection point onto the set of zeros of the operator. Our quantitative analysis, guided by Fernando Ferreira and Paulo Oliva’s bounded functional interpretation, provides a primitive recursive bound on the metastability for the convergence of the algorithm, in the sense of Terence Tao. Furthermore, we obtain quantitative information on the asymptotic regularity of the iteration. The results of this paper are made possible by an arithmetization of the lim sup.

Keywords: Convex optimization, proximal point algorithm, proof mining, metastability

Dinis Bruno, Pinto Pedro: Metastability of the proximal point algorithm with multi-parameters. Port. Math. 77 (2020), 345-381. doi: 10.4171/PM/2054