一类带变换${\ell }_1$罚函数的正则化问题的强度量次正则研究

Study on the Strong Metric Subregularity for a Class of Regularization Problems with Transformed ${\ell }_1$ Penalty Function

摘要: 本文研究一类带变换ℓ1罚函数的正则化问题，该模型的目标函数由损失函数和变换ℓ1罚函数两部分组成，其中损失函数是二次可微函数，变换ℓ1罚函数是一个非凸函数。本文首先研究变换ℓ1罚函数的邻近次微分和极限次微分，然后利用变换ℓ1罚函数的次微分表达式和集值映射的图像导数工具得到了该类问题目标函数的次微分的图像导数，最后利用该图像导数表达式分别建立了该正则化问题的强度量次正则的一个充分条件和充要条件。

Abstract: This paper studies the regularization problem for a class of penalty functions with transformedℓ1. The objective function of the model consists of two parts: the loss function and the transformedℓ1penalty function, where the loss function is a quadratic differentiable function and the transformedℓ1penalty function is a nonconvex function. This article first studies the proximal subdifferentials and limiting subdifferentials of the transformedℓ1penalty function. Then, by the subdifferential expression of the transformedℓ1penalty function and graphical derivative tool for set-value mapping, we obtain the graphical derivative of the subdifferentials of the objective function. Finally, using the graphical derivative expression, we establish a sufficient condition, a necessary and sufficient condition of the strong metric subregularity for the regularization problem.