AUGMENTED LAGRANGIAN WITHOUT ALTERNATING DIRECTIONS: PRACTICAL ALGORITHMS FOR INVERSE PROBLEMS IN IMAGING

Rahul Mourya (1) , Loic Denis (1) , Éric Thiébaut (2) , Jean-Marie Becker (1)

1 LHC - Laboratoire Hubert Curien
2 CRAL - Centre de Recherche Astrophysique de Lyon

Several problems in signal processing and machine learning can be casted as optimization problems. In many cases, they are of large-scale, nonlinear, have constraints, and may be nonsmooth in the unknown parameters. There exists plethora of fast algorithms for smooth convex optimization, but these algorithms are not readily applicable to nonsmooth problems , which has led to a considerable amount of research in this direction. In this paper, we propose a general algorithm for nonsmooth bound-constrained convex optimization problems. Our algorithm is instance of the so-called augmented Lagrangian, for which theoretical convergence is well established for convex problems. The proposed algorithm is a blend of superlinearly convergent limited memory quasi-Newton method, and proximal projection operator. The initial promising numerical results for total-variation based image deblurring show that they are as fast as the best existing algorithms in the same class, but with fewer and less sensitive tuning parameters, which makes a huge difference in practice.

Mots clés: total variation; deblurring; proximal operator; ADMM; hierarchical optimization; non-smooth optimization; Index Terms— Constrained convex optimization

Domaines: Traitement du signal et de l'image [eess.SP]; Optique [physics.optics]

Rahul Mourya, Loic Denis, Éric Thiébaut, Jean-Marie Becker. AUGMENTED LAGRANGIAN WITHOUT ALTERNATING DIRECTIONS: PRACTICAL ALGORITHMS FOR INVERSE PROBLEMS IN IMAGING. IEEE International Conference on Image Processing (ICIP), Sep 2015, Quebec, Canada. ⟨ujm-01122878⟩