Alternating Proximal Gradient Method for Convex Minimization

In this paper, we apply the idea of alternating proximal gradient to solve separable convex minimization problems with three or more blocks of variables linked by some linear constraints. The method proposed in this paper is to firstly group the variables into two blocks, and then apply a proximal gradient based inexact alternating direction method of multipliers to solve the new formulation. The main computational effort in each iteration of the proposed method is to compute the proximal mappings of the involved convex functions. The global convergence result of the proposed method is established. We show that many interesting problems arising from machine learning, statistics, medical imaging and computer vision can be solved by the proposed method. Numerical results on problems such as latent variable graphical model selection, stable principal component pursuit and compressive principal component pursuit are presented.