  This paper is concerned with the numerical minimization of energy functionals
in Hilbert spaces involving convex constraints coinciding with a semi-norm for
a subspace. The optimization is realized by alternating minimizations of the
functional on a sequence of orthogonal subspaces. On each subspace an iterative
proximity-map algorithm is implemented via \emph{oblique thresholding}, which
is the main new tool introduced in this work. We provide convergence conditions
for the algorithm in order to compute minimizers of the target energy.
Analogous results are derived for a parallel variant of the algorithm.
Applications are presented in domain decomposition methods for singular
elliptic PDE's arising in total variation minimization and in accelerated
sparse recovery algorithms based on $\ell_1$-minimization. We include numerical
examples which show efficient solutions to classical problems in signal and
image processing.
