  We introduce an iterative optimization scheme for convex objectives
consisting of a linear loss and a non-separable penalty, based on the
expectation-consistent approximation and the vector approximate message-passing
(VAMP) algorithm. Specifically, the penalties we approach are convex on a
linear transformation of the variable to be determined, a notable example being
total variation (TV). We describe the connection between message-passing
algorithms -- typically used for approximate inference -- and proximal methods
for optimization, and show that our scheme is, as VAMP, similar in nature to
the Peaceman-Rachford splitting, with the important difference that stepsizes
are set adaptively. Finally, we benchmark the performance of our VAMP-like
iteration in problems where TV penalties are useful, namely classification in
task fMRI and reconstruction in tomography, and show faster convergence than
that of state-of-the-art approaches such as FISTA and ADMM in most settings.
