An Extended Alternating Direction Method for Three-Block Separable Convex Programming

In order to solve the convex minimization problem with at least two variables, the augmented Lagrangian method is often used and improved to the alternating direction method of multipliers (ADMM) in the Gauss-Seidel or Jacobian fashion. Though various versions of the ADMM were developed for solving the two-block separable convex problem with linear equality constraint, there are a few feasible ways to tackling the problem with three-block objectives. In this paper, we design a novel ADMM combining with these two manners, where the first two subproblems are solved in parallel with some positive-definite proximal terms and the step sizes of updating the Lagrangian multipliers are enlarged. The variational inequality is applied to characterize the solution set of the concerned problem, and the Cauchy-Schwarz inequality is used to analyze some properties of the sequence {w−w̃} in a weighted norm, where w and w̃ are respectively called the predictive variable and correcting variable. In analyzing the convergence of the proposed method, the lower bound of ∥w− w̃∥G is discussed separately in several different cases. Then, the global convergence of the proposed method is proved and the worst-case O(1/t) convergence rate in an ergodic sense is established.