Inexact accelerated augmented Lagrangian methods

Abstract

The augmented Lagrangian method (ALM) is a popular method for solving linearly constrained convex minimization problems, and it has been used in many applications such as compressive sensing or image processing. Recently, accelerated versions of the augmented Lagrangian method (AALM) have been developed, and they assume that the subproblem can be exactly solved. However, the subproblem of the augmented Lagrangian method in general does not have a closed-form solution. In this paper, we introduce an inexact version of an accelerated augmented Lagrangian method (I-AALM), with an implementable inexact stopping condition for the subproblem. It is also proved that the convergence rate of our method remains the same as the accelerated ALM, which is $${{\mathcal {O}}}(\frac{1}{k^2})$$O(1k2) with an iteration number $$k$$k. In a similar manner, we propose an inexact accelerated alternating direction method of multiplier (I-AADMM), which is an inexact version of an accelerated ADMM. Numerical applications to compressive sensing or image inpainting are also presented to validate the effectiveness of the proposed iterative algorithms.