One to beat them all: "RYU" -- a unifying framework for the construction of safe balls

In this paper, we present a new framework, called "RYU" for constructing "safe" regions -- specifically, bounded sets that are guaranteed to contain the dual solution of a target optimization problem. Our framework applies to the standard case where the objective function is composed of two components: a closed, proper, convex function with Lipschitz-smooth gradient and another closed, proper, convex function. We show that the RYU framework not only encompasses but also improves upon the state-of-the-art methods proposed over the past decade for this class of optimization problems.