Geodesic
Ubiquitous Algorithms in Convex Optimization Generate Self-Contracted Sequences
Journal of convex analysis, Tome 29 (2022) no. 1, pp. 119-128
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We show that various algorithms, ubiquitous in convex optimization (e.g. pro\-ximal-gradient, alternating projections and averaged projections) generate self-con\-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.
Classification : 52A41, 65K05, 52A05, 90C25
Mots-clés : Proximal-gradient algorithm, alternating projection, self-contracted curve 
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     author = {A. B\"ohm and A. Daniilidis},
     title = {Ubiquitous {Algorithms} in {Convex} {Optimization} {Generate} {Self-Contracted} {Sequences}},
     journal = {Journal of convex analysis},
     pages = {119--128},
     year = {2022},
     volume = {29},
     number = {1},
     url = {http://geodesic.mathdoc.fr/item/JCA_2022_29_1_JCA_2022_29_1_a5/}
}
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A. Böhm; A. Daniilidis. Ubiquitous Algorithms in Convex Optimization Generate Self-Contracted Sequences. Journal of convex analysis, Tome 29 (2022) no. 1, pp. 119-128. http://geodesic.mathdoc.fr/item/JCA_2022_29_1_JCA_2022_29_1_a5/