对称的自身价值问题的地质凸度和最的下降的收
Foivos Alimisis1, Bart Vandereycken1
1Department of Mathematics, University of Geneva, Geneva, Switzerland.
Journal of optimization theory and applications
|November 4, 2024
概括
我们引入弱强凸性,以优化格拉斯曼变量上的雷利分数. 这使得最的下降算法能够更快地进行收分析,即使没有强大的凸度假设.
科学领域:
- 优化理论 优化理论
- 数字分析 数字分析
- 线性代数 线性代数
- 多重优化多重优化
背景情况:
- 在格拉斯曼多元体上,区块雷利分数最小化是主要元件分析等领域的一个关键问题.
- 这个问题在欧几里德空间中是非凸的,在里曼倍数上只有局部凸的,这给优化带来了挑战.
- 现有的方法通常依赖于强烈的凸度假设或eigengap条件,限制了它们的适用性.
研究的目的:
- 为了分析利曼最的下降算法的汇聚,以最小化区块雷利分数.
- 建立融合的新理论条件,放松传统的凸度要求.
- 为了更深入地了解格拉斯曼多元体上的优化问题的几何性质.
主要方法:
- 引入一种名为"弱强凸"的新概念来描述问题的结构.
- 将凸优化技术适应于利曼设置,使用弱强凸性属性.
- 在不同条件下分析里曼的最的下降算法的收率.
主要成果:
- 证明弱强凸性允许进行类似于凸问题的收分析.
- 当矩阵具有零 eigengap 时指数收率的证明,而不需要特定的初始化.
- 对其他情况的代数收率和在全球最小化器附近的地质凸度的证明.
结论:
- 弱强凸性框架提供了一个强大的工具,用于分析在变形体上的非凸优化问题.
- 里曼的最的下降算法对区块雷利分数问题表现出强大的收性质.
- 这项工作放松了初始化条件,并扩展了在多重优化中对汇率证明的适用性.
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