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Within the context of sequential methods for solving general nonlinear programming problems, and on the grounds of a previous work of the same authors, this study deals with the theoretical reasoning behind handling the original subproblems by an augmentation strategy. We do not assume feasibility of the original problem, nor the fulfillment of any constraint qualification. The previous analysis is extended along two directions. First and foremost, the exact nature of the stationary points previously considered is alleviated under an approximate stationary perspective. Second, the current analysis has been developed using general vector norms. Therefore, despite the similarities of the obtained results with those of the prior study, the present ones have been obtained under less restrictive hypotheses, and with a more involved examination. As before, we are not concerned with the sequential method itself, nor with computational results. We focus on the features of the original problem that can be inferred from the properties of the solution of the augmented problem, with the solutions being now analyzed in an approximate sense. Examples illustrating the obtained results are included.

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. This paper addresses the theoretical reasoning behind handling the original subproblems by an augmentation strategy related to the differentiable reformulation of the $$\ell _1$$ ℓ1 -penalized problem. Nevertheless, this paper is not concerned with the sequential method itself, but with the features about the original problem that can be inferred from the properties of the solution of the augmented problem. Moreover, no assumption is made upon the feasibility of the original problem, neither about the fulfillment of any constraint qualification, nor of any regularity condition, such as calmness. The convergence analysis of the involved sequences is presented, independent of the strategy employed to produce the iterates. Examples that elucidate the interrelations among the obtained results are also provided.

In this work we propose a derivative-free trust-region algorithm for unconstrained optimization. The algorithm is based on the ideas of Powell (Comput Optim Appl 53:527–555, 2012). We turn his general ideas into a step-form algorithm that avoids unnecessary reductions of the trust-region radius. The results related to the global convergence of the algorithm are presented in detail. We prove that, under reasonable hypotheses, any accumulation point of the sequence generated by the algorithm is stationary.