An efficient partial parallel method with scaling step size strategy for three-block convex optimization problems

A popular optimization model arising from image processing is the separable optimization problem whose objective function is the sum of three independent functions, and the variables are coupled by a linear equality. In this paper, we propose to solve such problem by a new partially parallel splitting method, whose step sizes for the primal and the dual variables in correction step are not necessarily identical. We establish the global convergence, and study the convergence rate on this varying ADMM-based prediction-correction method named as VAPCM. We derive the worst-case O(1/t) convergence rate in both the ergodic and non-ergodic senses. We also show that the convergence rate can be improved to o(1/t). Moreover, under the error bound assumptions, we establish the global linear convergence of VAPCM. We apply the new method to solve problems in robust principal component analysis and image decomposition. Numerical results indicate that the new method is efficient.