
Journal of Convex Analysis 29 (2022), No. 1, 119128 Copyright Heldermann Verlag 2022 Ubiquitous Algorithms in Convex Optimization Generate SelfContracted Sequences Axel Böhm Faculty of Mathematics, University of Vienna, 1090 Vienna, Austria axel.boehm@univie.ac.at Aris Daniilidis DIMCMM, UMI CNRS 2807, FCFM, Universidad de Chile, Santiago, Chile arisd@dim.uchile.cl [Abstractpdf] We show that various algorithms, ubiquitous in convex optimization (e.g. pro\ximalgradient, alternating projections and averaged projections) generate selfcon\trac\ted sequences $\{x_{k}\}_{k\in\mathbb{N}}$. As a consequence, a novel universal bound for the \emph{length} \ $\sum_{k\ge 0}\Vert x_{k+1}x_k\Vert$ \ can be deduced. In addition, this bound is independent of both the concrete data of the problem (sets, functions) as well as the stepsize involved, and only depends on the dimension of the space. Keywords: Proximalgradient algorithm, alternating projection, selfcontracted curve. MSC: 52A41, 65K05; 52A05, 90C25. [ Fulltextpdf (111 KB)] for subscribers only. 