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An Augmented Lagrangian Method for Quasi-Equilibrium Problems

Luis Felipe Bueno(lfelipebueno***at***gmail.com)
Gabriel Haeser(ghaeser***at***ime.usp.br)
Felipe Lara(felipelaraobreque***at***gmail.com)
Frank Navarro Rojas(franknr***at***ime.usp.br)

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.

Keywords: Augmented Lagrangian methods; Quasi-equilibrium problems; Equilibrium problems; Constraint qualifications; Approximate-KKT conditions.

Category 1: Nonlinear Optimization

Citation:

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Entry Submitted: 01/18/2019
Entry Accepted: 01/18/2019
Entry Last Modified: 01/18/2019

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