Modified hybrid decomposition of the augmented Lagrangian method with larger step size for three-block separable convex programming.

The Jacobian decomposition and the Gauss-Seidel decomposition of augmented Lagrangian method (ALM) are two popular methods for separable convex programming. However, their convergence is not guaranteed for three-block separable convex programming. In this paper, we present a modified hybrid decomposition of ALM (MHD-ALM) for three-block separable convex programming, which first updates all variables by a hybrid decomposition of ALM, and then corrects the output by a correction step with constant step size $\alpha \in (0,2-\sqrt{2})$ which is much less restricted than the step sizes in similar methods. Furthermore, we show that $2-\sqrt{2}$ is the optimal upper bound of the constant step size α. The rationality of MHD-ALM is testified by theoretical analysis, including global convergence, ergodic convergence rate, nonergodic convergence rate, and refined ergodic convergence rate. MHD-ALM is applied to solve video background extraction problem, and numerical results indicate that it is numerically reliable and requires less computation.