The Douglas–Rachford algorithm for a hyperplane and a doubleton
| Autor: | Heinz H. Bauschke, Scott B. Lindstrom, Minh N. Dao |
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| Rok vydání: | 2019 |
| Předmět: |
021103 operations research
Control and Optimization Applied Mathematics Mathematics::Optimization and Control 0211 other engineering and technologies Regular polygon 02 engineering and technology Management Science and Operations Research Characterization (mathematics) Computer Science Applications Set (abstract data type) 47H10 49M27 65K05 65K10 90C26 Hyperplane Optimization and Control (math.OC) Simple (abstract algebra) FOS: Mathematics Focus (optics) Mathematics - Optimization and Control Algorithm Mathematics |
| Zdroj: | Journal of Global Optimization. 74:79-93 |
| ISSN: | 1573-2916 0925-5001 |
| DOI: | 10.1007/s10898-019-00744-7 |
| Popis: | The Douglas--Rachford algorithm is a popular algorithm for solving both convex and nonconvex feasibility problems. While its behaviour is settled in the convex inconsistent case, the general nonconvex inconsistent case is far from being fully understood. In this paper, we focus on the most simple nonconvex inconsistent case: when one set is a hyperplane and the other a doubleton (i.e., a two-point set). We present a characterization of cycling in this case which --- somewhat surprisingly --- depends on whether the ratio of the distance of the points to the hyperplane is rational or not. Furthermore, we provide closed-form expressions as well as several concrete examples which illustrate the dynamical richness of this algorithm. |
| Databáze: | OpenAIRE |
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