Towards optimization techniques on diffeological spaces by generalizing Riemannian concepts
Publication date
2025-11-24
Document type
Forschungsartikel
Author
Organisational unit
Publisher
Springer
Series or journal
Applied Mathematics & Optimization
ISSN
Periodical volume
93
Periodical issue
1
Article ID
5
Peer-reviewed
✅
Part of the university bibliography
✅
Language
English
Abstract
Diffeological spaces firstly introduced by J. M. Souriau in the 1980 s are a natural generalization of smooth manifolds but optimization techniques are only known on manifolds so far. Generalizing these techniques to diffeological spaces is very challenging because of several reasons. One of the main reasons is that there are various definitions of tangent spaces which do not coincide. Additionally, one needs to deal with a generalization of a Riemannian space in order to define gradients which are indispensable for optimization methods. One main aim of this paper is a suitable definition of a tangent space in view to optimization methods. Based on this definition, we present a diffeological Riemannian space and a diffeological gradient, which we need for the formulation of an optimization algorithm on diffeological spaces. Moreover, in order to be able to update the iterates in an optimization algorithm on diffeological spaces, we present a diffeological retraction and the Levi-Civita connection on diffeological spaces. This paper also illustrates the novel objects by examples. Finally, we formulate the steepest descent method on diffeological spaces and apply it to an example.
Description
This article is licensed under a Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/).
Version
Published version
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Metadata only access
Open Access Funding
Springer Nature (DEAL)