Many applications arising in a variety of fields can be well illustrated by the task of recovering the low-rank and sparse components of a given matrix. Recently, it is discovered that this NP-hard task can be well accomplished, both theoretically and numerically, via heuristically solving a convex relaxation problem where the widely-acknowledged nuclear norm and l 1 norm are utilized to induce low-rank and sparsity. In the literature, it is conventionally assumed that all entries of the matrix to be recovered are exactly known (via observation). To capture even more applications, this paper studies the recovery task in more general settings: only a fraction of entries of the matrix can be observed and the observation is corrupted by both impulsive and Gaussian noise. The resulted model falls into the applicable scope of the classical augmented Lagrangian method. Moreover, the separable structure of the new model enables us to solve the involved subproblems more efficiently by splitting the augmented Lagrangian function. Hence, some implementable numerical algorithms are developed in the spirits of the well-known alternating direction method and the parallel splitting augmented Lagrangian method. Some preliminary numerical experiments verify that these augmented-Lagrangian-based algorithms are easily-implementable and surprisingly-efficient for tackling the new recovery model.

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Abstract. The total variation (TV) model is attractive for being able to preserve sharp attributes in images. However, the restored images from TV-based methods do not usually stay in a given dynamic range, and hence projection is required to bring them back into the dynamic range for visual presentation or for storage in digital media. This will affect the accuracy of the restoration as the projected image will no longer be the minimizer of the given TV model. In this paper, we show that one can get much more accurate solutions by imposing box constraints on the TV models and solving the resulting constrained models. Our numerical results show that for some images where there are many pixels with values lying on the boundary of the dynamic range, the gain can be as great as 10.28dB in peak signal-to-noise ratio. One traditional hinderance of using the constrained model is that it is difficult to solve. However, in this paper, we propose to use the alternating direction method of multipliers (ADMM) to solve the constrained models. This leads to a fast and convergent algorithm that is applicable for both Gaussian and impulse noise. Numerical results show that our ADMM algorithm is better than some state-of-the-art algorithms for unconstrained models both in terms of accuracy and robustness with respect to the regularization parameter.

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Abstract. The proximal point algorithm (PPA) has been well studied in the literature. In particular, its linear convergence rate has been studied by Rockafellar in 1976 under certain condition. We consider a generalized PPA in the generic setting of finding a zero point of a maximal monotone operator, and show that the condition proposed by Rockafellar can also sufficiently ensure the linear convergence rate for this generalized PPA. Indeed we show that these linear convergence rates are optimal.Both the exact and inexact versions of this generalized PPA are discussed. The motivation to consider this generalized PPA is that it includes as special cases the relaxed versions of some splitting methods that are originated from PPA. Thus, linear convergence results of this generalized PPA can be used to better understand the convergence of some widely used algorithms in the literature. We focus on the particular convex minimization context and specify Rockafellar's condition to see how to ensure the linear convergence rate for some efficient numerical schemes, including the classical augmented Lagrangian method proposed by Hensen and Powell in 1969 and its relaxed version, the original alternating direction method of multipliers (ADMM) by Glowinski and Marrocco in 1975 and its relaxed version (i.e., the generalized ADMM by Eckstein and Bertsekas in 1992).Some refined conditions weaker than existing ones are proposed in these particular contexts.

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Recently, in He et al. [He BS, Tao M, Yuan XM (2012) Alternating direction method with Gaussian back substitution for separable convex programming. SIAM J. Optim. 22(2):313–340], we have showed the first possibility of combining the Douglas-Rachford alternating direction method of multipliers (ADMM) with a Gaussian back substitution procedure for solving a convex minimization model with a general separable structure. This paper is a further study on this theme. We first derive a general algorithmic framework to combine ADMM with either a forward or backward substitution procedure. Then, we show that convergence of this framework can be easily proved from the contraction perspective, and its local linear convergence rate is provable if certain error bound condition is assumed. Without such an error bound assumption, we can estimate its worst-case convergence rate measured by the iteration complexity.

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