Sparse optimization problems in fractional order Sobolev spaces

We consider optimization problems in the fractional order Sobolev spaces \(H^s(\Omega)\), \(s\in (0,1)\), with sparsity promoting objective functionals containing \(L^p\)-pseudonorms, \(p\in (0,1)\). Existence of solutions is proven. By means of a smoothing scheme, we obtain first-order optimality c...

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Bibliographic Details
Published in:arXiv.org 2023-06
Main Authors: Antil, Harbir, Wachsmuth, Daniel
Format: Article
Language:English
Subjects:
Algorithms
Optimization
Smoothing
Sobolev space
Online Access:Get full text
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Summary:We consider optimization problems in the fractional order Sobolev spaces \(H^s(\Omega)\), \(s\in (0,1)\), with sparsity promoting objective functionals containing \(L^p\)-pseudonorms, \(p\in (0,1)\). Existence of solutions is proven. By means of a smoothing scheme, we obtain first-order optimality conditions. An algorithm based on this smoothing scheme is developed. Weak limit points of iterates are shown to satisfy a stationarity system that is slightly weaker than that given by the necessary condition.
ISSN:2331-8422
DOI:10.48550/arxiv.2204.11456