  The EM algorithm is a widely used methodology for penalized likelihood
estimation. Provable monotonicity and convergence are the hallmarks of the EM
algorithm and these properties are well established for smooth likelihood and
smooth penalty functions. However, many relaxed versions of variable selection
penalties are not smooth. The goal of this paper is to introduce a new class of
Space Alternating Penalized Kullback Proximal extensions of the EM algorithm
for nonsmooth likelihood inference. We show that the cluster points of the new
method are stationary points even when on the boundary of the parameter set.
Special attention has been paid to the construction of component-wise version
of the method in order to ease the implementation for complicated models.
Illustration for the problems of model selection for finite mixtures of
regression and to sparse image reconstruction is presented.
