  The recent literature on first order methods for smooth optimization shows
that significant improvements on the practical convergence behaviour can be
achieved with variable stepsize and scaling for the gradient, making this class
of algorithms attractive for a variety of relevant applications. In this paper
we introduce a variable metric in the context of the $\epsilon$-subgradient
projection methods for nonsmooth, constrained, convex problems, in combination
with two different stepsize selection strategies. We develop the theoretical
convergence analysis of the proposed approach and we also discuss practical
implementation issues, as the choice of the scaling matrix. In order to
illustrate the effectiveness of the method, we consider a specific problem in
the image restoration framework and we numerically evaluate the effects of a
variable scaling and of the steplength selection strategy on the convergence
behaviour.
