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

Space alternating penalized Kullback proximal point algorithms for maximizing likelihood with nondifferentiable penalty