The Largest-$K$-Norm for General Measure Spaces and a DC Reformulation for $L^0$-Constrained Problems in Function Spaces

Bastian Dittrich; Daniel Wachsmuth

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arXiv · 2403.19437

The Largest-$K$-Norm for General Measure Spaces and a DC Reformulation for $L^0$-Constrained Problems in Function Spaces

Bastian Dittrich·Daniel Wachsmuth

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Abstract

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.

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Bastian Dittrich, Daniel Wachsmuth. 2024-03-28. The Largest-$K$-Norm for General Measure Spaces and a DC Reformulation for $L^0$-Constrained Problems in Function Spaces. https://arxiv.org/abs/2403.19437

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