Convergence rate of alternating projection method for the intersection of an affine subspace and the second-order cone
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| Format: | Preprint |
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2024
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| author | Ochiai, Hiroyuki Sekiguchi, Yoshiyuki Waki, Hayato |
| author_facet | Ochiai, Hiroyuki Sekiguchi, Yoshiyuki Waki, Hayato |
| contents | We study the convergence rate of the alternating projection method (APM) applied to the intersection of an affine subspace and the second-order cone. We show that when they intersect non-transversally, the convergence rate is $O(k^{-1/2})$, where $k$ is the number of iterations of the APM. In particular, when the intersection is not at the origin or forms a half-line with the origin as the endpoint, the obtained convergence rate can be exact because a lower bound of the convergence rate is evaluated. These results coincide with the worst-case convergence rate obtained from the error bound discussed in [Borwein et al., SIOPT, 2014] and [Drusvyatskiy et al., Math. Prog., 2017]. Moreover, we consider the convergence rate of the APM for the intersection of an affine subspace and the product of two second-order cones. We provide an example that the worst-case convergence rate of the APM is better than the rate expected from the error bound for the example. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2401_02084 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Convergence rate of alternating projection method for the intersection of an affine subspace and the second-order cone Ochiai, Hiroyuki Sekiguchi, Yoshiyuki Waki, Hayato Optimization and Control Primary 41A25, 90C25, Secondary 65K10 We study the convergence rate of the alternating projection method (APM) applied to the intersection of an affine subspace and the second-order cone. We show that when they intersect non-transversally, the convergence rate is $O(k^{-1/2})$, where $k$ is the number of iterations of the APM. In particular, when the intersection is not at the origin or forms a half-line with the origin as the endpoint, the obtained convergence rate can be exact because a lower bound of the convergence rate is evaluated. These results coincide with the worst-case convergence rate obtained from the error bound discussed in [Borwein et al., SIOPT, 2014] and [Drusvyatskiy et al., Math. Prog., 2017]. Moreover, we consider the convergence rate of the APM for the intersection of an affine subspace and the product of two second-order cones. We provide an example that the worst-case convergence rate of the APM is better than the rate expected from the error bound for the example. |
| title | Convergence rate of alternating projection method for the intersection of an affine subspace and the second-order cone |
| topic | Optimization and Control Primary 41A25, 90C25, Secondary 65K10 |
| url | https://arxiv.org/abs/2401.02084 |