Appl Math Optim 53:259–277 (2006)
2006 Springer Science+Business Media, Inc.
Stability of the Minimizers of Least Squares
with a Non-Convex Regularization.
Part II: Global Behavior
and M. Nikolova
LAMFA UMR 6140, Universit´e de Picardie,
33 rue Saint-Leu, 90039 Amien Cedex, France
CMLA UMR 8536, ENS de Cachan,
61 av. du President Wilson, 94235 Cachan Cedex, France
Abstract. We address estimation problems where the sought-after solution is de-
ﬁned as the minimizer of an objective function composed of a quadratic data-ﬁdelity
term and a regularization term. We especially focus on non-convex and possibly non-
smooth regularization terms because of their ability to yield good estimates. This
work is dedicated to the stability of the minimizers of such piecewise C
m ≥ 2, non-convex objective functions. It is composed of two parts. In the previous
part of this work we considered general local minimizers. In this part we derive
results on global minimizers. We show that the data domain contains an open, dense
subset such that for every data point therein, the objective function has a ﬁnite num-
ber of local minimizers, and a unique global minimizer. It gives rise to a global
minimizer function which is C
everywhere on an open and dense subset of the
Key Words. Stability analysis, Regularized least squares, Non-smooth analysis,
Non-convex analysis, Signal and image processing.
AMS Classiﬁcation. 26B, 49J, 68U, 94A.
This is the second part of a work devoted to the stability of minimizers of regularized
least squares objective functions as customarily used in signal and image reconstruction.