Motivated by big data applications, first-order methods have been extremely popular in recent years. However, naive gradient methods generally converge slowly. Hence, much efforts have been made to accelerate various first-order methods. This paper proposes two accelerated methods towards solving structured linearly constrained convex programming, for which we assume composite convex objective that is the sum of a differentiable function and a possibly nondifferentiable one. The first method is the accelerated linearized augmented Lagrangian method (LALM). At each update to the primal variable, it allows linearization to the differentiable function and also the augmented term, and thus it enables easy subproblems. Assuming merely convexity, we show that LALM owns O(1/t) convergence if parameters are kept fixed during all the iterations and can be accelerated to O(1/t^2) if the parameters are adapted, where t is the number of total iterations. The second method is the accelerated linearized alternating direction method of multipliers (LADMM). In addition to the composite convexity, it further assumes two-block structure on the objective. Different from classic ADMM, our method allows linearization to the objective and also augmented term to make the update simple. Assuming strong convexity on one block variable, we show that LADMM also enjoys O(1=t2) convergence with adaptive parameters. This result is a significant improvement over that in [Goldstein et. al, SIIMS'14], which requires strong convexity on both block variables and no linearization to the objective or augmented term. Numerical experiments are performed on quadratic programming, image denoising, and support vector machine. The proposed accelerated methods are compared to nonaccelerated ones and also existing accelerated methods. The results demonstrate the validness of acceleration and superior performance of the proposed methods over existing ones.

The alternating direction method of multipliers (ADMM) is a popular and efficient first-order method that has recently found numerous applications, and the proximal ADMM is an important variant of it. The main contributions of this paper are the proposition and the analysis of a class of inertial proximal ADMMs, which unify the basic ideas of the inertial proximal point method and the proximal ADMM, for linearly constrained separable convex optimization. This class of methods are of inertial nature because at each iteration the proximal ADMM is applied to a point extrapolated at the current iterate in the direction of last movement. The recently proposed inertial primal-dual algorithm [A. Chambolle and T. Pock, On the ergodic convergence rates of a first-order primaldual algorithm, preprint, 2014, Algorithm 3] and the inertial linearized ADMM [C. Chen, S. Ma, and J. Yang, arXiv:1407.8238, eq. (3.23)] are covered as special cases. The proposed algorithmic framework is very general in the sense that the weighting matrices in the proximal terms are allowed to be only positive semidefinite, but not necessarily positive definite as required by existing methods of the same kind. By setting the two proximal terms to zero, we obtain an inertial variant of the classical ADMM, which is to the best of our knowledge new. We carry out a unified analysis for the entire class of methods under very mild assumptions. In particular, convergence, as well as asymptotic $o(1/\sqrt{k})$ and nonasymptotic $O(1/\sqrt{k})$ rates of convergence, are established for the best primal function value and feasibility residues, where k denotes the iteration counter. The global iterate convergence of the generated sequence is established under an additional assumption. We also present extensive experimental results on total variation–based image reconstruction problems to illustrate the profits gained by introducing the inertial extrapolation steps.

We consider image deblurring problem in the presence of impulsive noise. It is known that \emph{total variation} (TV) regularization with L1-norm penalized data fitting (TVL1 for short) works reasonably well only when the level of impulsive noise is relatively low. For high level impulsive noise, TVL1 works poorly. The reason is that all data, both corrupted and noise free, are equally penalized in data fitting, leading to insurmountable difficulty in balancing regularization and data fitting. In this paper, we propose to combine TV regularization with nonconvex \emph{smoothly clipped absolute deviation} (SCAD) penalty for data fitting (TVSCAD for short). Our motivation is simply that data fitting should be enforced only when an observed data is not severely corrupted, while for those data more likely to be severely corrupted, less or even null penalization should be enforced. A \emph{difference of convex functions} algorithm is adopted to solve the nonconvex TVSCAD model, resulting in solving a sequence of TVL1-equivalent problems, each of which can then be solved efficiently by the alternating direction method of multipliers. Theoretically, we establish global convergence to a critical point of the nonconvex objective function. The R-linear and at-least-sublinear convergence rate results are derived for the cases of anisotropic and isotropic TV, respectively. Numerically, experimental results are given to show that the TVSCAD approach improves those of the TVL1 significantly, especially for cases with high level impulsive noise, and is comparable with the recently proposed iteratively corrected TVL1 method [\textit{Inverse Problems}, 32 (2016), pp.~085004].