矩阵的谱算子
Spectral Operators of Matrices
摘要：
近年来，矩阵最优化问题（MOPs）已被公认为是一种强有力的工具，被研究者们远远超越了优化界，用来对涉及结构低秩矩阵的许多重要应用进行建模。这种趋势在一定程度上可以归功于新兴的压缩传感领域令人兴奋的发展。Löwner算子通过对矩阵的每个奇异值应用单变量函数来生成矩阵值函数，长期以来在解决矩阵最优化问题中发挥着重要作用。然而，为Löwner算子开发的经典理论在这些最近的应用中已经变得不足。本文的主要目的是为设计求解多目标优化问题的数值方法提供必要的理论基础。通过引入一类新的矩阵值函数，并对其进行深入的研究来实现这一目标，这类函数被称为矩阵的谱算子。系统地研究了谱算子的几个基本性质，包括良定性、连续性、方向可微性、Fréchet可微性、局部利普希茨连续性、$ρ$阶B（ouligand）可微性（$0<ρ\leq 1$）、$ρ$阶G-半光滑性（$0<ρ\leq 1$）以及Clarke广义雅可比矩阵的刻画.
英文摘要：
The class of matrix optimization problems (MOPs) has been recognized in recent years to be a powerful tool by researchers far beyond the optimization community to model many important applications involving structured low rank matrices. This trend can be credited to some extent to the exciting developments in the emerging field of compressed sensing. The Löwner operator, which generates a matrix valued function by applying a single-variable function to each of the singular values of a matrix, has played an important role for a long time in solving matrix optimization problems. However, the classical theory developed for Löwner operators has become inadequate in these recent applications. The main objective of this paper is to provide some necessary theoretical foundations for designing numerical methods for solving the MOP. This goal is achieved by introducing and conducting a thorough study on a new class of matrix valued functions, coined as spectral operators of matrices. Several fundamental properties of spectral operators, including the well-definedness, continuity, directional differentiability, Fréchet-differentiability, locally Lipschitzian continuity, $ρ$-order B(ouligand)-differentiability ($0<ρ\leq 1$), $ρ$-order G-semismooth ($0<ρ\leq 1$) and the characterization of Clarke's generalized Jacobian, are systematically studied.
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