交替同时的Halpern-Lions-Wittmann-Bauschke算法用于找到凸集合的两个不连接的交叉点的最佳近似对
Yair Censor1, Rafiq Mansour1, Daniel Reem1
1Department of Mathematics, University of Haifa, Mt. Carmel, Haifa 3498838, Israel.
Journal of approximation theory
|June 10, 2024
概括
本研究引入了一种代投影方法,以找到两个不连接的凸集之间的最近点. 这种新的方法避免了直接投射到交叉点上,简化了计算并扩展了以前的发现.
科学领域:
- 凸的分析 凸的分析
- 优化理论 优化理论
- 功能分析是一种功能分析.
背景情况:
- 在优化中,在不连接的凸集之间找到最接近的点至关重要.
- 现有的方法通常需要对交叉点进行投影,这可能是计算密集的.
- 在Aharoni,Censor和Jiang之前的工作中,他们讨论了有限维度的多面体.
研究的目的:
- 开发一种新的代算法,用于在封闭凸子集的两个不连接的交点之间找到最佳近似对.
- 为了克服直接投射到交叉点上的计算负担.
- 将现有结果扩展到更一般的环境中.
主要方法:
- 基于对生成子集 (而不是交叉点) 的投影的代过程.
- 该算法受到Halpern-Lions-Wittmann-Bauschke和Cheney-Goldstein交替过程的影响.
- 收是在特定条件下被证明的.
主要成果:
- 建议的代过程汇聚到一个最佳近似对.
- 当底层空间是欧几里德空间,生成子集是紧且严格凸起时,收就得到了保证.
- 该方法避免了对交叉点本身进行计算要求高的投影的需要.
结论:
- 新的代投影算法提供了一种有效的方式,可以在不连接的凸集合之间找到最佳近似对.
- 在重要的实用案例中,收的条件是满足的,比如Euclidean空间与紧的,严格凸起的生成集.
- 这项工作对多面体集合的先前结果进行了概括和改进.
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