稀疏加组稀疏优化问题的一阶和二阶方向稳定点研究

On the First-Order and Second-Order Directional Stationary Points of Sparse Plus Group Sparse Optimization Problems

摘要: 在本文中，我们考虑一类非凸非光滑的无约束稀疏加组稀疏优化问题，其损失函数是二阶连续可微函数(可能非凸)，惩罚项是稀疏惩罚与组稀疏惩罚的组合，其稀疏惩罚是ℓ1范数，组稀疏惩罚是折叠凹惩罚函数。目前，计算这类带有凸加非凸惩罚优化问题的方向稳定点的研究较少，但利用方向导数定义的方向稳定点比次微分所定义的稳定点(critical点、lifted稳定点等)能更好的刻画解的局部最优性质。因此，本文主要通过方向稳定点来刻画模型的最优性条件。首先，本文引入了一阶、二阶方向稳定点的概念，探讨了它们与问题局部解的关系。其次，给出了一阶、二阶方向导数的具体表达式，这为进一步分析和求解此类问题提供了理论基础。

Abstract: In this paper, we consider a class of nonconvex, nonsmooth and unconstrained sparse plus group sparse optimization problems, in which the loss function is a twice continuously differentiable function (possibly nonconvex), and the penalty term is a combination of sparse penalty and group sparse penalty, where the sparse penalty is a ℓ1 norm, and the group sparse penalty is a general folded concave penalty function. At present, there are few studies on the calculation of directional stationary points for this type of optimization problems with convex plus nonconvex penalties, and directional stationary points characterized by directional derivatives can better show the local optimal properties of solutions than other stationary points defined by subdifferentials (e.g. critical points, lifted stationary points, etc.). Therefore, in this paper, the optimal-ity conditions of the model are characterized by means of directional stationary points. Firstly, we introduce the concepts of first-order and second-order directional stationary points, and discuss their relations with local solutions of the problem. Secondly, the concrete expressions of the first and second order directional derivatives are given, which provide a theoretical basis for further analyzing and solving this kind of problems.