We present a numerical iterative optimization algorithm for the minimization
of a cost function consisting of a linear combination of three convex terms,
one of which is differentiable, a second one is prox-simple and the third one
is the composition of a linear map and a prox-simple function. The algorithm's
special feature lies in its ability to approximate, in a single iteration run,
the minimizers of the cost function for many different values of the parameters
determining the relative weight of the three terms in the cost function. A
proof of convergence of the algorithm, based on an inexact variable metric
approach, is also provided. As a special case, one recovers a generalization of
the primal-dual algorithm of Chambolle and Pock, and also of the
proximal-gradient algorithm. Finally, we show how it is related to a
primal-dual iterative algorithm based on inexact proximal evaluations of the
non-smooth terms of the cost function.Comment: 22 pages, 2 figure