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Recently, the efficient solvers for compressive sensing (CS) problems with Total Variation (TV) regularization are needed, mainly because of the reconstruction of an image by a single pixel camera, or the recovery of a medical image from its partial Fourier samples. In this paper, we propose an alternating directions scheme algorithm for solving the TV regularized minimization problems with linear constraints. We minimize the corresponding augmented Lagrangian function alternatively at each step. Both of the resulting subproblems admit explicit solutions by applying a linear-time shrinkage. The algorithm is easily performed, in which, only two matrix-vector multiplications and two fast Fourier transforms are involved at per-iteration. The global convergence of the proposed algorithm follows directly in this literature. Numerical comparisons with the sate-of-theart method TVLA3 illustrate that the proposed method is effective and promising.

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In this paper, we propose a generalized alternating direction method of multipliers (ADMM) with semi-proximal terms for solving a class of convex composite conic optimization problems, of which some are high-dimensional, to moderate accuracy. Our primary motivation is that this method, together with properly chosen semi-proximal terms, such as those generated by the recent advance of block symmetric Gauss-Seidel technique, is capable of tackling these problems. Moreover, the proposed method, which relaxes both the primal and the dual variables in a natural way with a common relaxation factor in the interval of (0, 2), has the potential of enhancing the performance of the classic ADMM. Extensive numerical experiments on various doubly non-negative semidefinite programming problems, with or without inequality constraints, are conducted. The corresponding results showed that all these multi-block problems can be successively solved, and the advantage of using the relaxation step is apparent.

Total variation (TV) regularization is popular in image reconstruction due to its edge-preserving property. In this paper, we extend the alternating minimization algorithm recently proposed in [37] to the case of recovering images from random projections. Specifically, we propose to solve the TV regularized least squares problem by alternating minimization algorithms based on the classical quadratic penalty technique and alternating minimization of the augmented Lagrangian function. The per-iteration cost of the proposed algorithms is dominated by two matrix-vector multiplications and two fast Fourier transforms. Convergence results, including finite convergence of certain variables and q-linear convergence rate, are established for the quadratic penalty method. Furthermore, we compare numerically the new algorithms with some state-of-the-art algorithms. Our experimental results indicate that the new algorithms are stable, efficient and competitive with the compared ones.

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