  In this paper, we put forth a novel framework (named ``RYU'') for the
construction of ``safe'' balls, i.e. regions that provably contain the dual
solution of a target optimization problem. We concentrate on the standard setup
where the cost function is the sum of two terms: a closed, proper, convex
Lipschitz-smooth function and a closed, proper, convex function. The RYU
framework is shown to generalize or improve upon all the results proposed in
the last decade for the considered family of optimization problems.
