Trade-off Invariance Principle for minimizers of regularized functionals
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arXiv
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| Format: | Preprint |
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2024
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| author | Fornasier, Massimo Klemenc, Jona Scagliotti, Alessandro |
| author_facet | Fornasier, Massimo Klemenc, Jona Scagliotti, Alessandro |
| contents | In this paper, we consider functionals of the form $H_α(u)=F(u)+αG(u)$ with $α\in[0,+\infty)$, where $u$ varies in a set $U\neq\emptyset$ (without further structure). We first revisit a result stating that, excluding at most countably many values of $α$, we have $\inf_{H_α^\star}G= \sup_{H_α^\star}G$, where $H_α^\star := \arg\min_UH_α$, which is assumed to be non-empty. Then, we prove a stronger result that concerns the invariance of the limiting value of the functional $G$ along minimizing sequences for $H_α$, which extends the above Principle to the case $H_α^\star= \emptyset$. Moreover, we show to what extent these findings generalize to multi-regularized functionals and -- in the presence of an underlying differentiable structure -- to critical points. Finally, the main result implies an unexpected consequence for functionals regularized with uniformly convex norms: excluding again at most countably many values of $α$, it turns out that for a minimizing sequence, convergence to a minimizer in the weak or strong sense is equivalent. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_11639 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Trade-off Invariance Principle for minimizers of regularized functionals Fornasier, Massimo Klemenc, Jona Scagliotti, Alessandro Optimization and Control In this paper, we consider functionals of the form $H_α(u)=F(u)+αG(u)$ with $α\in[0,+\infty)$, where $u$ varies in a set $U\neq\emptyset$ (without further structure). We first revisit a result stating that, excluding at most countably many values of $α$, we have $\inf_{H_α^\star}G= \sup_{H_α^\star}G$, where $H_α^\star := \arg\min_UH_α$, which is assumed to be non-empty. Then, we prove a stronger result that concerns the invariance of the limiting value of the functional $G$ along minimizing sequences for $H_α$, which extends the above Principle to the case $H_α^\star= \emptyset$. Moreover, we show to what extent these findings generalize to multi-regularized functionals and -- in the presence of an underlying differentiable structure -- to critical points. Finally, the main result implies an unexpected consequence for functionals regularized with uniformly convex norms: excluding again at most countably many values of $α$, it turns out that for a minimizing sequence, convergence to a minimizer in the weak or strong sense is equivalent. |
| title | Trade-off Invariance Principle for minimizers of regularized functionals |
| topic | Optimization and Control |
| url | https://arxiv.org/abs/2411.11639 |