Finite Convergence of Circumcentered-Reflection Method on Closed Polyhedral Cones in Euclidean Spaces
Fuente:
arXiv
Guardado en:
| Autor principal: | |
|---|---|
| Formato: | Preprint |
| Publicado: |
2024
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866915042336702464 |
|---|---|
| author | Liao, Hongzhi |
| author_facet | Liao, Hongzhi |
| contents | The Circumcentered Reflection Method (CRM) is a recently developed projection method for solving convex feasibility problems. It offers preferable convergence properties compared to classic methods such as the Douglas-Rachford and the alternating projections method. In this study, our first main theorem establishes that CRM can identify a feasible point in the intersection of two closed convex cones in \(\mathbb{R}^2\) from any starting point in the Euclidean plane. We then apply this theorem to intersections of two polyhedral sets in \(\mathbb{R}^2\) and two wedge-like sets in \(\mathbb{R}^n\), proving that CRM converges to a point in the intersection from any initial position finitely. Additionally, we introduce a modified technique based on CRM, called the Sphere-Centered Reflection Method. With the help of this technique, we demonstrate that CRM can locate a feasible point in finitely many iterations in the intersection of two proper polyhedral cones in \(\mathbb{R}^3\) when the initial point lies in a subset of the complement of the intersection's polar cone. Lastly, we provide an example illustrating that finite convergence may fail for the intersection of two proper polyhedral cones in \(\mathbb{R}^3\) if the initial guess is outside the designated set. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_00512 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Finite Convergence of Circumcentered-Reflection Method on Closed Polyhedral Cones in Euclidean Spaces Liao, Hongzhi Optimization and Control The Circumcentered Reflection Method (CRM) is a recently developed projection method for solving convex feasibility problems. It offers preferable convergence properties compared to classic methods such as the Douglas-Rachford and the alternating projections method. In this study, our first main theorem establishes that CRM can identify a feasible point in the intersection of two closed convex cones in \(\mathbb{R}^2\) from any starting point in the Euclidean plane. We then apply this theorem to intersections of two polyhedral sets in \(\mathbb{R}^2\) and two wedge-like sets in \(\mathbb{R}^n\), proving that CRM converges to a point in the intersection from any initial position finitely. Additionally, we introduce a modified technique based on CRM, called the Sphere-Centered Reflection Method. With the help of this technique, we demonstrate that CRM can locate a feasible point in finitely many iterations in the intersection of two proper polyhedral cones in \(\mathbb{R}^3\) when the initial point lies in a subset of the complement of the intersection's polar cone. Lastly, we provide an example illustrating that finite convergence may fail for the intersection of two proper polyhedral cones in \(\mathbb{R}^3\) if the initial guess is outside the designated set. |
| title | Finite Convergence of Circumcentered-Reflection Method on Closed Polyhedral Cones in Euclidean Spaces |
| topic | Optimization and Control |
| url | https://arxiv.org/abs/2412.00512 |