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On the best achievable quality of limit points of augmented Lagrangian schemes

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Abstract

The optimization literature is vast in papers dealing with improvements on the global convergence of augmented Lagrangian schemes. Usually, the results are based on weak constraint qualifications, or, more recently, on sequential optimality conditions obtained via penalization techniques. In this paper, we propose a somewhat different approach, in the sense that the algorithm itself is used in order to formulate a new optimality condition satisfied by its feasible limit points. With this tool at hand, we present several new properties and insights on limit points of augmented Lagrangian schemes, in particular, characterizing the strongest possible global convergence result for the safeguarded augmented Lagrangian method.

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Acknowledgements

We would like to thank the reviewers of this paper for their many insightful suggestions.

Funding

This work has been partially supported by CEPID-CeMEAI (FAPESP 2013/07375-0), FAPES (grant 116/2019), FAPESP (grants 2018/24293-0, 2017/18308-2 and 2017/17840-2), CNPq (grants 301888/2017-5, 303427/2018-3, 404656/2018-8, 438185/2018-8 and 307270/2019-0), and PRONEX - CNPq/FAPERJ (grant E-26/010.001247/2016).

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Authors and Affiliations

  1. Department of Applied Mathematics, University of Campinas, Rua Sérgio Buarque de Holanda, 651, 13083-859, Campinas, SP, Brazil

    Roberto Andreani

  2. Department of Applied Mathematics, University of São Paulo, Rua do Matão, 1010, Cidade Universitária, 05508-090, São Paulo, SP, Brazil

    Gabriel Haeser & Leonardo M. Mito

  3. Department of Mathematics, Federal University of Paraná, 81531-980, Curitiba, PR, Brazil

    Alberto Ramos

  4. Department of Applied Mathematics, Federal University of Espírito Santo, Rodovia BR 101, Km 60, 29932-540, São Mateus, ES, Brazil

    Leonardo D. Secchin

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  1. Roberto Andreani
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  5. Leonardo D. Secchin
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Correspondence to Roberto Andreani.

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Andreani, R., Haeser, G., Mito, L.M. et al. On the best achievable quality of limit points of augmented Lagrangian schemes. Numer Algor 90, 851–877 (2022). https://doi.org/10.1007/s11075-021-01212-8

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