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On the Fulfillment of the Complementary Approximate Karush–Kuhn–Tucker Conditions and Algorithmic Applications

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Abstract

Focusing on smooth constrained optimization problems, and inspired by the complementary approximate Karush–Kuhn–Tucker (CAKKT) conditions, this work introduces the weighted complementary approximate Karush–Kuhn–Tucker (WCAKKT) conditions. They are shown to be verified by limit points generated not only by safeguarded augmented Lagrangian methods, but also by inexact restoration methods, inverse and logarithmic barrier methods, and a penalized algorithm for constrained nonsmooth optimization. Under the analyticity of the feasible set description, and resting upon a desingularization result, the new conditions are proved to be equivalent to the CAKKT conditions. The WCAKKT conditions capture the algebraic elements of the desingularization result needed to characterize CAKKT sequences using a weighted complementarity condition that asymptotically sums zero. Due to its generality and strength, the new condition may help to enlighten the practical performance of algorithms in generating CAKKT sequences.

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Data Availability Statement

Data sharing does not apply to this article as no datasets were generated or analyzed during the current theoretical study.

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Acknowledgements

This work was partially supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) grant 305010/2020-4, and Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) Grants 2013/07375-0, 2016/22989-2, 2018/24293-0, 2019/18859-4. The authors are thankful to André Belotto da Silva for the fruitful discussions regarding desingularization and monomialization, as well as to the editor and the reviewers, for the constructive comments that improved the presentation of our work.

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  1. University of Campinas, Campinas, São Paulo, Brazil

    Renan W. Prado, Sandra A. Santos & Lucas E. A. Simões

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  1. Renan W. Prado
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Prado, R.W., Santos, S.A. & Simões, L.E.A. On the Fulfillment of the Complementary Approximate Karush–Kuhn–Tucker Conditions and Algorithmic Applications. J Optim Theory Appl 197, 705–736 (2023). https://doi.org/10.1007/s10957-023-02189-1

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