Convergence Analysis Of The Alternating Proximal Gradient Method And It’s Application

Multi-block convex optimization problems have wide applications in compressed sensing,computer image processing, singal processing, multi-task learning and so on. Many practicalproblems in matrix complexity, image recovery and machine learning can be abstracted into theproblem of minimizing the sum of several convex functions.In this paper, we consider a class of multi-block convex optimization problems where theobjective function is the composition of several(N≥3) convex functions, and the proximalmappings of the convex functions are easy. Because of the non-smoothness of the objectivefunctions, this class of convex optimization problems cannot be solved by traditional methodsdirectly. However, by proximity operator, algorithms can overcome these difficulties. The al-ternating proximal gradient method in this paper is an effective method to solve such problems,and the global convergence result is obtained.This article mainly study the convergence rate of the alternating proximal gradient method.From the perspective of variational inequalities, we establish the sub-linear convergence of itunder the assumption that the coefficient matrices are orthogonal. And in this paper, we usethe alternating proximal gradient method to solve a special class of unconstrained convex op-timization problems. In the last, we show the specific solution process and the correspondingalgorithm to solve the problem.