Expansions of the Characteristic Polynomial of a Perturbed Positive Semidefinite Matrix and Convergence Analysis of Alternating Projections
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866912330045980672 |
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| author | Ochiai, Hiroyuki Sekiguchi, Yoshiyuki Waki, Hayato |
| author_facet | Ochiai, Hiroyuki Sekiguchi, Yoshiyuki Waki, Hayato |
| contents | We observe that the characteristic polynomial of a linearly perturbed semidefinite matrix can be used to give the convergence rate of alternating projections for the positive semidefinite cone and a line. As a consequence, we show that such alternating projections converge at $O(k^{-1/2})$, independently of the singularity degree. A sufficient condition for the linear convergence is also obtained. Our method directly analyzes the defining equation for an alternating projection sequence and does not use error bounds. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_16689 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Expansions of the Characteristic Polynomial of a Perturbed Positive Semidefinite Matrix and Convergence Analysis of Alternating Projections Ochiai, Hiroyuki Sekiguchi, Yoshiyuki Waki, Hayato Optimization and Control Primary 90C25, 41A25, Secondary 65K10 We observe that the characteristic polynomial of a linearly perturbed semidefinite matrix can be used to give the convergence rate of alternating projections for the positive semidefinite cone and a line. As a consequence, we show that such alternating projections converge at $O(k^{-1/2})$, independently of the singularity degree. A sufficient condition for the linear convergence is also obtained. Our method directly analyzes the defining equation for an alternating projection sequence and does not use error bounds. |
| title | Expansions of the Characteristic Polynomial of a Perturbed Positive Semidefinite Matrix and Convergence Analysis of Alternating Projections |
| topic | Optimization and Control Primary 90C25, 41A25, Secondary 65K10 |
| url | https://arxiv.org/abs/2401.16689 |