Abstract
In this paper, we propose an Augmented Lagrangian algorithm for solving a general class of possible non-convex problems called quasi-equilibrium problems (QEPs). We define an Augmented Lagrangian bifunction associated with QEPs, introduce a secondary QEP as a measure of infeasibility and we discuss several special classes of QEPs within our theoretical framework. For obtaining global convergence under a new weak constraint qualification, we extend the notion of an Approximate Karush–Kuhn–Tucker (AKKT) point for QEPs (AKKT-QEP), showing that in general it is not necessarily satisfied at a solution, differently from its counterpart in optimization. We study some particular cases where AKKT-QEP does hold at a solution, while discussing the solvability of the subproblems of the algorithm. We also present illustrative numerical experiments.
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Acknowledgements
We would like to express gratitude to the anonymous referees whose valuable suggestions improved this paper. This work was supported by FAPESP (2013/05475-7, 2018/24293-0 and 2017/18308-2), CAPES and CNPq. For F. Lara, this research was partially supported by Conicyt–Chile under project Fondecyt Iniciación 11180320.
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Bueno, L.F., Haeser, G., Lara, F. et al. An Augmented Lagrangian method for quasi-equilibrium problems. Comput Optim Appl 76, 737–766 (2020). https://doi.org/10.1007/s10589-020-00180-4
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DOI: https://doi.org/10.1007/s10589-020-00180-4