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.

中文翻译：

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拟均衡问题的增强拉格朗日方法

在本文中，我们提出了一种增强的拉格朗日算法，用于求解一类可能的非凸问题，称为拟均衡问题（QEP）。我们定义了与QEP相关的增强型Lagrangian双功能，引入了次级QEP作为不可行的度量，并且我们在理论框架内讨论了几类特殊的QEP。为了在新的弱约束条件下获得全局收敛，我们扩展了QEP的近似Karush–Kuhn–Tucker（AKKT）点（AKKT-QEP）的概念，这表明与解决方案不同的是，不一定总能满足其对应的优化。在讨论算法子问题的可解性的同时，我们研究了AKKT-QEP确实可以解决的一些特殊情况。我们还提出了说明性的数值实验。