A new separable property of the joint numerical range of quadratic functions and its applications to the Smallest Enclosing Ball Problem
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866909854861361152 |
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| author | Nguyen, Van-Bong Nguyen, Huu-Quang |
| author_facet | Nguyen, Van-Bong Nguyen, Huu-Quang |
| contents | We explore separable property of the joint numerical range $G(\Bbb R^n)$ of a special class of quadratic functions and apply it to solving the smallest enclosing ball (SEB) problem which asks to find a ball $B(a,r)$ in $\Bbb R^n$ with smallest radius $r$ such that $B(a,r)$ contains the intersection $\cap_{i=1}^mB(a_i,r_i)$ of $m$ given balls $B(a_i,r_i).$ We show that $G(\Bbb R^n)$ is convex if and only if ${\rm rank}\{a_1-a, a_2-a, \ldots, a_m-a\}\le n-1.$ Otherwise, ${\rm rank}\{a_1-a, a_2-a, \ldots, a_m-a\}=n$ and $G(\Bbb R^n)$ is not convex. In this case we propose a new set $G(\Bbb R^n)^\bullet$ which allows to show that if $m=n$ then $G(\Bbb R^n)^\bullet$ is convex even $G(\Bbb R^n)$ is not. Importantly, the separable property of $G(\Bbb R^n)^\bullet$ then implies the separable property for $G(\Bbb R^n).$ As a result, a new progress on solving the SEB problem is obtained. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_15399 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A new separable property of the joint numerical range of quadratic functions and its applications to the Smallest Enclosing Ball Problem Nguyen, Van-Bong Nguyen, Huu-Quang Optimization and Control We explore separable property of the joint numerical range $G(\Bbb R^n)$ of a special class of quadratic functions and apply it to solving the smallest enclosing ball (SEB) problem which asks to find a ball $B(a,r)$ in $\Bbb R^n$ with smallest radius $r$ such that $B(a,r)$ contains the intersection $\cap_{i=1}^mB(a_i,r_i)$ of $m$ given balls $B(a_i,r_i).$ We show that $G(\Bbb R^n)$ is convex if and only if ${\rm rank}\{a_1-a, a_2-a, \ldots, a_m-a\}\le n-1.$ Otherwise, ${\rm rank}\{a_1-a, a_2-a, \ldots, a_m-a\}=n$ and $G(\Bbb R^n)$ is not convex. In this case we propose a new set $G(\Bbb R^n)^\bullet$ which allows to show that if $m=n$ then $G(\Bbb R^n)^\bullet$ is convex even $G(\Bbb R^n)$ is not. Importantly, the separable property of $G(\Bbb R^n)^\bullet$ then implies the separable property for $G(\Bbb R^n).$ As a result, a new progress on solving the SEB problem is obtained. |
| title | A new separable property of the joint numerical range of quadratic functions and its applications to the Smallest Enclosing Ball Problem |
| topic | Optimization and Control |
| url | https://arxiv.org/abs/2501.15399 |