An Augmented Lagrangian method for quasi-equilibrium problems

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...

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Bibliographic Details
Published in:	Computational optimization and applications Vol. 76; no. 3; pp. 737 - 766
Main Authors:	Bueno, L. F., Haeser, G., Lara, F., Rojas, F. N.
Format:	Journal Article
Language:	English
Published:	New York Springer US 01-07-2020 Springer Nature B.V
Subjects:	Algorithms Convex and Discrete Geometry Management Science Mathematics Mathematics and Statistics Operations Research Operations Research/Decision Theory Optimization Statistics Quasi-equilibrium problems Augmented Lagrangian methods Approximate-KKT conditions Equilibrium problems Constraint qualifications
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Summary:	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.
ISSN:	0926-6003 1573-2894
DOI:	10.1007/s10589-020-00180-4