  The importance of an adequate inner loop starting point (as opposed to a
sufficient inner loop stopping rule) is discussed in the context of a numerical
optimization algorithm consisting of nested primal-dual proximal-gradient
iterations. While the number of inner iterations is fixed in advance,
convergence of the whole algorithm is still guaranteed by virtue of a
warm-start strategy for the inner loop, showing that inner loop "starting
rules" can be just as effective as "stopping rules" for guaranteeing
convergence. The algorithm itself is applicable to the numerical solution of
convex optimization problems defined by the sum of a differentiable term and
two possibly non-differentiable terms. One of the latter terms should take the
form of the composition of a linear map and a proximable function, while the
differentiable term needs an accessible gradient. The algorithm reduces to the
classical proximal gradient algorithm in certain special cases and it also
generalizes other existing algorithms. In addition, under some conditions of
strong convexity, we show a linear rate of convergence.
