| Summary: | Abstract 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 α∈(0,2−2) $\alpha \in(0,2-\sqrt{2})$ which is much less restricted than the step sizes in similar methods. Furthermore, we show that 2−2 $2-\sqrt{2}$ 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.
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