  This paper introduces and studies the convergence properties of a new class
of explicit $\epsilon$-subgradient methods for the task of minimizing a convex
function over the set of minimizers of another convex minimization problem. The
general algorithm specializes to some important cases, such as first-order
methods applied to a varying objective function, which have computationally
cheap iterations. We present numerical experimentation regarding certain
applications where the theoretical framework encompasses efficient algorithmic
techniques, enabling the use of the resulting methods to solve very large
practical problems arising in tomographic image reconstruction.
