On Second-Order Cone Functions
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866909194290987008 |
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| author | Jibrin, Shafiu Swift, James W. |
| author_facet | Jibrin, Shafiu Swift, James W. |
| contents | We consider the second-order cone function (SOCF) $f: {\mathbb R}^n \to \mathbb R$ defined by $f(x)= c^T x + d -\|A x + b \|$. Every SOCF is concave. We give necessary and sufficient conditions for strict concavity of $f$. The parameters $A \in {\mathbb R}^{m \times n}$ and $b \in {\mathbb R}^m$ are not uniquely determined. We show that every SOCF can be written in the form $f(x) = c^T x + d -\sqrt{δ^2 + (x-x_*)^TM(x-x_*)}$. We give necessary and sufficient conditions for the parameters $c$, $d$, $δ$, $M = A^T A$, and $x_*$ to be uniquely determined. We also give necessary and sufficient conditions for $f$ to be bounded above. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_01360 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On Second-Order Cone Functions Jibrin, Shafiu Swift, James W. Optimization and Control 90C22, 90C25, 90C51 We consider the second-order cone function (SOCF) $f: {\mathbb R}^n \to \mathbb R$ defined by $f(x)= c^T x + d -\|A x + b \|$. Every SOCF is concave. We give necessary and sufficient conditions for strict concavity of $f$. The parameters $A \in {\mathbb R}^{m \times n}$ and $b \in {\mathbb R}^m$ are not uniquely determined. We show that every SOCF can be written in the form $f(x) = c^T x + d -\sqrt{δ^2 + (x-x_*)^TM(x-x_*)}$. We give necessary and sufficient conditions for the parameters $c$, $d$, $δ$, $M = A^T A$, and $x_*$ to be uniquely determined. We also give necessary and sufficient conditions for $f$ to be bounded above. |
| title | On Second-Order Cone Functions |
| topic | Optimization and Control 90C22, 90C25, 90C51 |
| url | https://arxiv.org/abs/2308.01360 |