Please use this identifier to cite or link to this item: https://hdl.handle.net/10419/307047 
Year of Publication: 
2022
Citation: 
[Journal:] Journal of Global Optimization [ISSN:] 1573-2916 [Volume:] 85 [Issue:] 3 [Publisher:] Springer US [Place:] New York, NY [Year:] 2022 [Pages:] 709-741
Publisher: 
Springer US, New York, NY
Abstract: 
Regularization is used in many different areas of optimization when solutions are sought which not only minimize a given function, but also possess a certain degree of regularity. Popular applications are image denoising, sparse regression and machine learning. Since the choice of the regularization parameter is crucial but often difficult, path-following methods are used to approximate the entire regularization path, i.e., the set of all possible solutions for all regularization parameters. Due to their nature, the development of these methods requires structural results about the regularization path. The goal of this article is to derive these results for the case of a smooth objective function which is penalized by a piecewise differentiable regularization term. We do this by treating regularization as a multiobjective optimization problem. Our results suggest that even in this general case, the regularization path is piecewise smooth. Moreover, our theory allows for a classification of the nonsmooth features that occur in between smooth parts. This is demonstrated in two applications, namely support-vector machines and exact penalty methods.
Subjects: 
Regularization
Nonsmooth analysis
Multiobjective optimization
Persistent Identifier of the first edition: 
Creative Commons License: 
cc-by Logo
Document Type: 
Article
Document Version: 
Published Version

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