  We study the \emph{Proximal Alternating Predictor-Corrector} (PAPC) algorithm
introduced recently by Drori, Sabach and Teboulle to solve nonsmooth structured
convex-concave saddle point problems consisting of the sum of a smooth convex
function, a finite collection of nonsmooth convex functions and bilinear terms.
We introduce the notion of pointwise quadratic supportability, which is a
relaxation of a standard strong convexity assumption and allows us to show that
the primal sequence is R-linearly convergent to an optimal solution and the
primal-dual sequence is globally Q-linearly convergent. We illustrate the
proposed method on total variation denoising problems and on locally adaptive
estimation in signal/image deconvolution and denoising with multiresolution
statistical constraints.
