In the context of sequential methods for solving general nonlinear programming problems, it is usual to work with augmented subproblems instead of the original ones, tackled by the $\ell_1$-penalty function together with the shortcut usage of a convenient penalty parameter. This paper addresses the theoretical reasoning behind handling the original subproblems by such an augmentation strategy, by means of the differentiable reformulation of the $\ell_1$-penalized problem. The convergence properties of related sequences of problems are analyzed. Furthermore, examples that elucidate the interrelations among the obtained results are presented.
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