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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2403.19437 |
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| _version_ | 1866912075744280576 |
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| author | Dittrich, Bastian Wachsmuth, Daniel |
| author_facet | Dittrich, Bastian Wachsmuth, Daniel |
| contents | We consider constraints on the measure of the support for integrable functions on arbitrary measure spaces. It is shown that this non-convex and discontinuous constraint can be equivalently reformulated by the difference of two convex and continuous functions, namely the $L^1$-norm and the so-called largest-$K$-norm. The largest-$K$-norm is studied and its convex subdifferential is derived. A corresponding penalty method is proposed, and its numerical solution by a DC method is investigated. Numerical experiments for two example problems, including a sparse optimal control problem, are presented. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_19437 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Largest-$K$-Norm for General Measure Spaces and a DC Reformulation for $L^0$-Constrained Problems in Function Spaces Dittrich, Bastian Wachsmuth, Daniel Optimization and Control 49K20, 49M20 We consider constraints on the measure of the support for integrable functions on arbitrary measure spaces. It is shown that this non-convex and discontinuous constraint can be equivalently reformulated by the difference of two convex and continuous functions, namely the $L^1$-norm and the so-called largest-$K$-norm. The largest-$K$-norm is studied and its convex subdifferential is derived. A corresponding penalty method is proposed, and its numerical solution by a DC method is investigated. Numerical experiments for two example problems, including a sparse optimal control problem, are presented. |
| title | The Largest-$K$-Norm for General Measure Spaces and a DC Reformulation for $L^0$-Constrained Problems in Function Spaces |
| topic | Optimization and Control 49K20, 49M20 |
| url | https://arxiv.org/abs/2403.19437 |