Abstract

Subdifferentials (in the sense of convex analysis) of matrix-valued functions defined on \(\mathbb {R}^d\) that are convex with respect to the Löwner partial order can have a complicated structure and might be very difficult to compute even in simple cases. The aim of this paper is to study subdifferential calculus for such functions and properties of their subdifferentials. We show that many standard results from convex analysis no longer hold true in the matrix-valued case. For example, in this case the subdifferential of the sum is not equal to the sum of subdifferentials, the Clarke subdifferential is not equal to the subdifferential in the sense of convex analysis, etc. Nonetheless, it is possible to provide simple rules for computing nonempty subsets of subdifferentials (in particular, individual subgradients) of convex matrix-valued functions in the general case and to completely describe subdifferentials of such functions defined on the real line. As a by-product of our analysis, we derive some interesting properties of convex matrix-valued functions, e.g. we show that if such function is nonsmooth, then its diagonal elements must be nonsmooth as well.

中文翻译：

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凸矩阵值函数的次微分

摘要

在\(\mathbb {R}^d\)上定义的矩阵值函数的次微分（在凸分析的意义上）相对于 Löwner 偏序是凸的，可能具有复杂的结构，甚至可能很难计算在简单的情况下。本文的目的是研究此类函数的次微分及其次微分的性质。我们表明，凸分析的许多标准结果在矩阵值情况下不再适用。例如，在这种情况下，和的次微分不等于次微分之和，克拉克次微分不等于凸分析意义上的次微分等。尽管如此，可以提供计算非空的简单规则一般情况下凸矩阵值函数的次微分子集（特别是各个次梯度），并完全描述在实线上定义的此类函数的次微分。作为我们分析的副产品，我们推导了凸矩阵值函数的一些有趣的性质，例如，我们表明，如果这样的函数是非光滑的，那么它的对角线元素也一定是非光滑的。