A multiobjective generalized Nash equilibrium problem (MGNEP) is a Nash equilibrium problem with constraints that include multiobjective games. We focus in this paper on examining the approximate Karush–Kuhn–Tucker (KKT) conditions for MGNEPs and their impact on the global convergence of algorithms. To begin, we define standard approximate weak KKT (standard-AWKKT) conditions for MGNEPs. We demonstrate that every weak efficient solution meets the standard-AWKKT condition. As these optimality conditions are strong, we propose a new set of approximate weak KKT conditions, specifically tailored for MGNEPs, and show their convergence towards weak KKT points provided a cone-continuity property is satisfied. It is important to note that while these newly introduced approximate weak KKT conditions do not serve as optimality conditions for general MGNEPs, we illustrate that they hold true for the special case of a weak variational equilibrium point in a jointly convex MGNEP. Finally, we propose an enhanced Lagrangian-type algorithm for estimating an approximate weak KKT of an MGNEP and demonstrate its global convergence.

中文翻译：

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多目标广义纳什均衡问题中的近似弱最优条件

多目标广义纳什均衡问题 (MGNEP) 是具有包括多目标博弈约束的纳什均衡问题。本文重点研究 MGNEP 的近似 Karush-Kuhn-Tucker (KKT) 条件及其对算法全局收敛的影响。首先，我们为 MGNEP 定义标准近似弱 KKT（标准 AWKKT）条件。我们证明每个弱有效解都满足标准 AWKKT 条件。由于这些最优性条件很强，我们提出了一组新的近似弱 KKT 条件，专门为 MGNEP 量身定制，并在满足锥连续性属性的情况下展示了它们向弱 KKT 点的收敛性。值得注意的是，虽然这些新引入的近似弱 KKT 条件不能作为一般 MGNEP 的最优条件，但我们说明它们对于联合凸 MGNEP 中弱变分平衡点的特殊情况成立。最后，我们提出了一种增强的拉格朗日型算法来估计 MGNEP 的近似弱 KKT 并证明了其全局收敛性。