Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)This paper introduces and studies the convergence properties of a new class of explicit e-subgradient methods for the task of minimizing a convex function over a 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 concerning 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.335FAPESP [2013/07375-0, 2013/16508-3, 2011/02219-4, 2013/14615-7]CNPq [311476/2014-7]Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)