On the Convexification of Spectral Sets Induced by Non-Invariant Sets
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866911466150428672 |
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| author | Zhao, Renbo |
| author_facet | Zhao, Renbo |
| contents | Given a finite-dimensional FTvN system $(\mathbb{V},\mathbb{W},λ)$, we study the convexification of the spectral set $λ^{-1}(\mathcal{C})$ induced by a set $\mathcal{C} \subseteq \mathbb{W}$. While the case of invariant $\mathcal{C}$ has been relatively well-studied, the results for non-invariant $\mathcal{C}$ are largely lacking in the literature. We fill this void by developing simple and geometric characterizations of the convex hull and closed convex hull of $λ^{-1}(\mathcal{C})$ when $\mathcal{C}$ has no invariance property. We further specialize our results to the case of invariant $\mathcal{C}$, and obtain new convexifications of $λ^{-1}(\mathcal{C})$ in this case. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_14143 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the Convexification of Spectral Sets Induced by Non-Invariant Sets Zhao, Renbo Optimization and Control Given a finite-dimensional FTvN system $(\mathbb{V},\mathbb{W},λ)$, we study the convexification of the spectral set $λ^{-1}(\mathcal{C})$ induced by a set $\mathcal{C} \subseteq \mathbb{W}$. While the case of invariant $\mathcal{C}$ has been relatively well-studied, the results for non-invariant $\mathcal{C}$ are largely lacking in the literature. We fill this void by developing simple and geometric characterizations of the convex hull and closed convex hull of $λ^{-1}(\mathcal{C})$ when $\mathcal{C}$ has no invariance property. We further specialize our results to the case of invariant $\mathcal{C}$, and obtain new convexifications of $λ^{-1}(\mathcal{C})$ in this case. |
| title | On the Convexification of Spectral Sets Induced by Non-Invariant Sets |
| topic | Optimization and Control |
| url | https://arxiv.org/abs/2405.14143 |