On generators of $k$-PSD closures of the positive semidefinite cone
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866914781154246656 |
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| author | Bhardwaj, Avinash Narayanan, Vishnu Pathapati, Abhishek |
| author_facet | Bhardwaj, Avinash Narayanan, Vishnu Pathapati, Abhishek |
| contents | Positive semidefinite (PSD) cone is the cone of positive semidefinite matrices, and is the object of interest in semidefinite programming (SDP). A computational efficient approximation of the PSD cone is the $k$-PSD closure, $1 \leq k < n$, cone of $n\times n$ real symmetric matrices such that all of their $k\times k$ principal submatrices are positive semidefinite. For $k=1$, one obtains a polyhedral approximation, while $k=2$ yields a second order conic (SOC) approximation of the PSD cone. These approximations of the PSD cone have been used extensively in real-world applications such as AC Optimal Power Flow (ACOPF) to address computational inefficiencies where SDP relaxations are utilized for convexification the non-convexities. However a theoretical discussion about the geometry of these conic approximations of the PSD cone is rather sparse. In this short communication, we attempt to provide a characterization of some family of generators of the aforementioned conic approximations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_01208 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On generators of $k$-PSD closures of the positive semidefinite cone Bhardwaj, Avinash Narayanan, Vishnu Pathapati, Abhishek Optimization and Control 90C22, 90C25, 52A27 Positive semidefinite (PSD) cone is the cone of positive semidefinite matrices, and is the object of interest in semidefinite programming (SDP). A computational efficient approximation of the PSD cone is the $k$-PSD closure, $1 \leq k < n$, cone of $n\times n$ real symmetric matrices such that all of their $k\times k$ principal submatrices are positive semidefinite. For $k=1$, one obtains a polyhedral approximation, while $k=2$ yields a second order conic (SOC) approximation of the PSD cone. These approximations of the PSD cone have been used extensively in real-world applications such as AC Optimal Power Flow (ACOPF) to address computational inefficiencies where SDP relaxations are utilized for convexification the non-convexities. However a theoretical discussion about the geometry of these conic approximations of the PSD cone is rather sparse. In this short communication, we attempt to provide a characterization of some family of generators of the aforementioned conic approximations. |
| title | On generators of $k$-PSD closures of the positive semidefinite cone |
| topic | Optimization and Control 90C22, 90C25, 52A27 |
| url | https://arxiv.org/abs/2405.01208 |