The lower-order regularization problem (i.e., the regularization problem (0 < p < 1)) has been widely studied for finding sparse solutions of linear inverse problems and gained successful applications in various mathematics and applied science fields. Several descent methods have been proposed and investigated for solving the regularization problem. However, a complete optimality condition of local minima and the convergence rates of the descent methods for the regularization problem have not been well investigated yet. To remedy this gap, in this talk, we will first provide a necessary and sufficient optimality condition for local minima and to show the property of strict local minima of order 2 for the regularization problem. This not only shows the advantage of applying regularization to induce sparse solutions, but also provides a tool for the linear convergence study of the descent methods. Then we will establish the linear convergence of the descent methods under the simple assumption that the limiting point of iterates is a local minimum of the regularization problem. Applying this abstract convergence theorem, we will also study the linear convergence of the well-known proximal gradient algorithm, as well as the inexact proximal gradient algorithm.

 

This work is joint work with Dr. Yaohua Hu (Shenzhen University), Dr. Kaiwen Meng (Southwest Jiaotong University) and Prof. Xiaoqi Yang (Hong Kong Polytechnic University).