Stability of Minimizers of Least Squares with a Non convex Regularization. Part I: Local Behavior

Sylvain Durand (1) , Mila Nikolova (2)

1 LAMFA - Laboratoire Amiénois de Mathématique Fondamentale et Appliquée
2 CMLA - Centre de Mathématiques et de Leurs Applications

Many estimation problems amount to minimizing an objective function composed of a quadratic data-fidelity term and a general regularization term. It is widely accepted that the minimizers obtained using nonsmooth and/or nonconvex regularization terms are frequently good estimates. However, vey few facts are known on the ways to control properties of these minimizers. This work is dedicated to the stability of the minimizers of such nonsmooth and/or nonconvex objective functions. It consists of two parts: in this part, we focus on general lical minimizers, whereas in a second part, we derive result on global minimizers. Her we demonstrate that the date domain contains an open, dense subset whose elements give rise to local and global minimizers which are necessaril strict. Moreover, we show that the relevant minimizers are stable under variations of the date

Mots clés: stability analysis; regularized least-squares; non-smoth nalysis; non-convex analysis; signal processing

Domaines: Optimisation et contrôle [math.OC]; Traitement du signal et de l'image [eess.SP]; Traitement du signal et de l'image [eess.SP]

Sylvain Durand, Mila Nikolova. Stability of Minimizers of Least Squares with a Non convex Regularization. Part I: Local Behavior. Applied Mathematics and Optimization, 2006, 53 (2), pp.185-208. ⟨10.1007/s00245-005-0842-1⟩. ⟨hal-00205023⟩