Abstract
The augmented Lagrangian method (ALM) is one of the most successful first-order methods for convex programming with linear equality constraints. To solve the two-block separable convex minimization problem, we always use the parallel splitting ALM method. In this paper, we will show that no matter how small the step size and the penalty parameter are, the convergence of the parallel splitting ALM is not guaranteed. We propose a new convergent parallel splitting ALM (PSALM), which is the regularizing ALM’s minimization subproblem by some simple proximal terms. In application this new PSALM is used to solve video background extraction problems and our numerical results indicate that this new PSALM is efficient.
Highlights
Many problems arising from machine learning, such as compressive sensing [1, 2], the video background extraction problem [3,4,5], batch images alignment [6, 7], and transform invariant low-rank textures [8, 9] can be formulated as separable convex programming with linear constraints
We demonstrate the potential efficiency of this new parallel splitting ALM (PSALM) by solving the video background extraction problem (6)
We have proposed a new parallel splitting augmented Lagrangian method(NEW− PSALM) for twoblock separable convex programming and have established its various convergence results, including global convergence, ergodic, and nonerogdic convergence rate
Summary
Many problems arising from machine learning, such as compressive sensing [1, 2], the video background extraction problem [3,4,5], batch images alignment [6, 7], and transform invariant low-rank textures [8, 9] can be formulated as separable convex programming with linear constraints. Focusing on the way of splitting the augmented Lagrangian function in the spirit of the well-known alternating direction method (ADM), Tao and Yuan [4] have proposed a variant of the alternating splitting augmented Lagrangian method (ASALM) with convergent property, which can solve three-block separable convex programming. He et al [13] propose a splitting method for solving a separable convex minimization problem with linear constraints, where the objective function is expressed as the sum of m individual functions without coupled variables.
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