  We propose a proximal approach to deal with a class of convex variational
problems involving nonlinear constraints. A large family of constraints, proven
to be effective in the solution of inverse problems, can be expressed as the
lower level set of a sum of convex functions evaluated over different, but
possibly overlapping, blocks of the signal. For such constraints, the
associated projection operator generally does not have a simple form. We
circumvent this difficulty by splitting the lower level set into as many
epigraphs as functions involved in the sum. A closed half-space constraint is
also enforced, in order to limit the sum of the introduced epigraphical
variables to the upper bound of the original lower level set. In this paper, we
focus on a family of constraints involving linear transforms of distance
functions to a convex set or $\ell_{1,p}$ norms with $p\in \{1,2,\infty\}$. In
these cases, the projection onto the epigraph of the involved function has a
closed form expression.
  The proposed approach is validated in the context of image restoration with
missing samples, by making use of constraints based on Non-Local Total
Variation. Experiments show that our method leads to significant improvements
in term of convergence speed over existing algorithms for solving similar
constrained problems. A second application to a pulse shape design problem is
provided in order to illustrate the flexibility of the proposed approach.
