An Efficient Augmented Lagrangian Method with Applications to Total Variation Minimization

Abstract

Based on the classic augmented Lagrangian multiplier method, we propose, analyze and test an algorithm for solving a class of equality-constrained non-smooth optimization problems (chiefly but not necessarily convex programs) with a particular structure. The algorithm effectively combines an alternating direction technique with a nonmonotone line search to minimize the augmented Lagrangian function at each iteration. We establish convergence for this algorithm, and apply it to solving problems in image reconstruction with total variation regularization. We present numerical results showing that the resulting solver, called TVAL3, is competitive with, and often outperforms, other state-of-the-art solvers in the field.

Efficient TpV minimization for circular, cone-beam computed tomography reconstruction via non-convex optimization

“…ADM framework is very suitable for solving the proposed model in that the proposed model has two separate variables. Moreover, the convergence analysis and conclusions [25] guarantee that this kind of methodology is of efficient performance. Note that (8) becomes conventional constrained TV minimization when p = 1.…”

“…With the aid of GPU accelerations of forward and backward projections [25], the time required for five and six projections are 1.1 and 1.2 s per iteration, respectively. In the first group of noiseless data simulations, six projections are used to perform the reconstruction.…”

A Strictly Contractive Peaceman--Rachford Splitting Method for Convex Programming

“…Our aim is to further verify the efficiency of the proposed strictly contractive PRSM (1.5) by comparing it numerically with four well-known algorithms in the imaging literature: SALSA [1], TwIST [4], SpaRSA [52], FISTA [2], and YALL1/TVAL3 [36, 37, 38, 54, 57]. …”

An linearized alternating direction method for total variation image restoration

“…Due to the non-differentiability and non-linearity of the TV term in problem (2), this problem is computationally challenging to solve despite its simple form. In this paper, we propose to solve TV deconvolution problems [ ] 10 , 4 , 3 by linearized alternating direction method (LADM)-a variant of the classic augmented Lagrangian method for structured optimization.…”