An Efficient Augmented Lagrangian Method with Semismooth Newton Solver for Total Generalized Variation

TL;DR

This paper introduces a semismooth Newton based augmented Lagrangian method for efficiently solving total generalized variation problems, demonstrating improved convergence and competitiveness over first-order methods.

Contribution

The paper develops a novel semismooth Newton augmented Lagrangian algorithm tailored for TGV, with proven global and local convergence properties.

Findings

The proposed method achieves faster convergence than existing first-order methods.

The algorithm demonstrates strong global and local linear convergence.

Numerical experiments confirm the efficiency and effectiveness of the approach.

Abstract

Total generalization variation (TGV) is a very powerful and important regularization for various inverse problems and computer vision tasks. In this paper, we proposed a semismooth Newton based augmented Lagrangian method to solve this problem. The augmented Lagrangian method (also called as method of multipliers) is widely used for lots of smooth or nonsmooth variational problems. However, its efficiency usually heavily depends on solving the coupled and nonlinear system together and simultaneously, which is very complicated and highly coupled for total generalization variation. With efficient primal-dual semismooth Newton methods for the complicated linear subproblems involving total generalized variation, we investigated a highly efficient and competitive algorithm compared to some efficient first-order method. With the analysis of the metric subregularities of the corresponding functions, we give both the global convergence and local linear convergence rate for the proposed augmented Lagrangian methods.