GLOBAL LINEAR CONVERGENCE RATE OF THE LINEARIZED VERSION OF THE GENERALIZED ALTERNATING DIRECTION METHOD OF MULTIPLIERS FOR SEPARABLE CONVEX PROGRAMMING

To solve the separable convex optimization problem with linear constraints, Fang et. al proposed the linearized version of generalized alternating direction method of multipliers (in short, L-GADMM) and the doubly proximal version of generalized alternating direction method of multipliers (in short, DPGADMM). Fang et. al also proved the worst-case O(l/t) convergence rate of both the (L-GADMM) and the (DL-GADMM) measured by the iteration complexity in both ergodic and nonergodic senses. In this paper, we establish the global linear convergence rate of both the L-GADMM and the DL-GADMM for separable convex optimization problem with the condition that the subdifferentials of the underlying functions are piecewise linear multifunctions. We also illustrate the effectiveness of the DL-GADMM by some numerical examples about calibrating the correlation matrices. The results in this paper extend, generalize and improve some known results in the literature.