仿射集上的低秩矩阵Optimization
Low-Rank Matrix Optimization Over Affine Set
摘要：
基于仿射集的低秩矩阵优化（秩-MOA）是指在一个与一个仿射集相交的低秩集上最小化一个连续可微函数。在适当的假设条件下，建立了Fréchet法锥与秩MOA可行集的相交规则。这使我们能够提出一阶最优性条件的秩MOA在所谓的F-平稳点和$α$ -平稳点。并给出了二阶最优性分析，包括必要条件和充分条件。这些结果丰富了低秩矩阵优化理论，为设计求解低秩矩阵优化问题的有效数值算法提供了潜在的线索。同时，我们说明了我们提出的秩MOA的几个具体应用，包括系统辨识中的Hankel矩阵逼近问题和信号处理中的线性流形上的低秩表示的最优性分析。
英文摘要：
The low-rank matrix optimization with affine set (rank-MOA) is to minimize a continuously differentiable function over a low-rank set intersecting with an affine set. Under some suitable assumptions, the intersection rule of the Fréchet normal cone to the feasible set of the rank-MOA is established. This allows us to propose the first-order optimality conditions for the rank-MOA in terms of the so-called F-stationary point and the $α$-stationary point. Furthermore, the second-order optimality analysis, including the necessary condition and the sufficient one, is proposed as well. 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, we illustrate our proposed optimality analysis for several specific applications of the rank-MOA including the Hankel matrix approximation problem in system identification and the low-rank representation on linear manifold in signal processing.
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