论文标题
正常锥体交点规则和低级矩阵优化的最佳分析与仿射歧管
Normal Cones Intersection Rule and Optimality Analysis for Low-Rank Matrix Optimization with Affine Manifolds
论文作者
论文摘要
具有仿射歧管（Rank-MOA）的低级矩阵优化旨在最大程度地减少在低级别集合与仿射歧管相交的连续可区分函数。本文致力于对等级-MOA的最佳分析。作为一个基石，在某些轻度的线性独立性假设下，建立了Fréchet正常锥体与可行的秩-MOA集合的相交规则。研究了基础正常锥的产生的显式公式，所谓的F-stationary Point和等级-MOA的α-平稳点，然后根据一阶最佳条件来揭示与局部/全局最小化器的关系。此外，根据模型的二阶分化信息，提出了二阶最佳分析，包括必要条件和足够的条件。所有这些结果将丰富低级基质优化的理论，并为设计有效的数值算法提供潜在的线索，以寻求低级解决方案。同时，讨论了排名-MOA的两个特定应用，以说明我们提出的最佳分析。
The low-rank matrix optimization with affine manifold (rank-MOA) aims to minimize a continuously differentiable function over a low-rank set intersecting with an affine manifold. This paper is devoted to the optimality analysis for rank-MOA. As a cornerstone, the intersection rule of the Fréchet normal cone to the feasible set of the rank-MOA is established under some mild linear independence assumptions. Aided with the resulting explicit formulae of the underlying normal cone, the so-called F-stationary point and the α-stationary point of rank-MOA are investigated and the relationship with local/global minimizers are then revealed in terms of first-order optimality conditions. Furthermore, the second-order optimality analysis, including the necessary and the sufficient conditions, is proposed based on the second-order differentiation information of the model. All these results will enrich the theory of low-rank matrix optimization and give potential clues to designing efficient numerical algorithms for seeking low rank solutions. Meanwhile, two specific applications of the rank-MOA are discussed to illustrate our proposed optimality analysis.
