On the approximate solutions of augmented subproblems within sequential methods for nonlinear programming

Abstract

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.