On the approximate solutions of augmented subproblems within sequential methods for nonlinear programming
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On the approximate solutions of augmented subproblems within sequential methods for nonlinear programming Ademir A. Ribeiro1 · Mael Sachine1 · Sandra A. Santos2 Received: 3 April 2018 / Revised: 6 August 2018 / Accepted: 22 August 2018 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2018
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. Keywords Nonlinear programming · KKT conditions · Approximate stationarity · Smooth reformulation · Sequential methods Mathematics Subject Classification 90C30 · 65K05 · 49M37
Communicated by Ernesto G. Birgin. This work was partially supported by FAPESP (Grants 2013/05475-7, 2013/07375-0), CNPq (Grants 302915/2016-8, 309437/2016-4, 442158/2014-9) and PRONEX-CNPq/FAPERJ.
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Ademir A. Ribeiro [email protected] Mael Sachine [email protected] Sandra A. Santos [email protected]
1
Federal University of Paraná, Curitiba, Brazil
2
IMECC-UNICAMP, University of Campinas, Campinas, Brazil
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A. A. Ribeiro et al.
1 Introduction Sequential techniques are commonly used for solving general nonlinear programming problems. Among such techniques, those based on models for the algebraic constraints, like successive linear programming (Fletcher and Sainz de la Maza 1989; Gomes and Senne 2011), sequential quadratic programming (SQP) (Gill et al. 2002), convex linearizations (CONLIN) (Fleury 1989), the method of moving asymptotes (MMA) (Svanberg 1987), as well as its extensions and modifications (Gomes-Ruggiero et al. 2010; Svanberg 2002; Zhang et al. 1996), might come up with unfeasible subproblems. Aiming at circumventing such a drawback, the addition of artificial variables and dealing with an augmented formulation may be convenient tools. In a previous work (Ribeiro et al. 2017), the authors have analyzed the theoretical reasoning behind handling the original subproblems by an augmented strategy, related to the differentia
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