An augmented Lagrangian method with exact multipliers for non-separable composite $\ell_0$-$\ell_2$ regularization
This paper studies a non-separable composite $\ell_0$-$\ell_2$ regularization model that simultaneously enforces sparsity and smoothness for inverse problems. The $\ell_0$ norm induces inherent nonconvexity and nonsmoothness, while linear transformations further introduce nonseparability, making the problem computationally challenging to solve. The existing inexact augmented Lagrangian method suffers from high computational complexity and unstable convergence. To overcome these difficulties, we develop two novel augmented Lagrangian algorithms with exact multipliers, designed respectively for the full row-rank case and the general matrix case, where all subproblems are globally optimized via closed-form solutions. Furthermore, we prove linear convergence of the proposed method when the transformation matrix is full row rank. In the general setting, all accumulation points of the generated sequence are KKT points for the original problem. Numerical experiments on synthetic data, trend filtering, and image smoothing demonstrate the superior efficiency and accuracy of the proposed methods over the existing method, confirming our theoretical analysis.