SIAM Journal on Optimization, Volume 31, Issue 1, Page 377-403, January 2021.
We consider the generalized Nash equilibrium problem (GNEP) with $ N $ players in a Hilbert space setting. The joint constraints are eliminated by an augmented Lagrangian-type approach, leading to an ADMM (alternating direction method of multipliers) algorithm. In contrast to standard optimization problems, however, the direct extension of ADMM to GNEPs is not necessarily convergent even for $N = 2$ players. We therefore use a regularized version of ADMM and present a global convergence result for $ N \geq 2 $ players under a partial strong monotonicity and a partial Lipschitz condition. Furthermore, also different from the optimization context, it turns out that the corresponding regularization parameters have to be sufficiently large in order to guarantee global convergence. We therefore also discuss a second ADMM-type method with an adaptive choice of the regularization parameters, with the aim of keeping the regularization parameters smaller and, hence, getting faster convergence. Numerical results are presented for some examples arising in infinite-dimensional Hilbert spaces.

中文翻译：

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Hilbert空间中广义Nash平衡问题的ADMM型方法

SIAM优化杂志，第31卷，第1期，第377-403页，2021年1月。
我们考虑希尔伯特空间环境中具有$ N $玩家的广义Nash均衡问题（GNEP）。通过增强的拉格朗日类型方法消除了联合约束，从而产生了ADMM（乘数的交替方向方法）算法。但是，与标准优化问题相反，即使对于$ N = 2 $的玩家，ADMM向GNEP的直接扩展也不一定会收敛。因此，我们使用正则化的ADMM版本，并在部分强单调性和部分Lipschitz条件下，为$ N \ geq 2 $玩家提供全局收敛结果。此外，与优化上下文不同的是，为了保证全局收敛，相应的正则化参数必须足够大。因此，我们还讨论了第二种ADMM类型的方法，该方法具有自适应选择的正则化参数，目的是使正则化参数保持较小并因此获得更快的收敛性。给出了在无穷维希尔伯特空间中产生的一些示例的数值结果。