We consider the problem of computing an equilibrium in a class of \textit{nonlinear generalized Nash equilibrium problems (NGNEPs)} in which the strategy sets for each player are defined by equality and inequality constraints that may depend on the choices of rival players. While the asymptotic global convergence and local convergence rates of algorithms to solve this problem have been extensively investigated, the analysis of nonasymptotic iteration complexity is still in its infancy. This paper presents two first-order algorithms -- based on the quadratic penalty method (QPM) and augmented Lagrangian method (ALM), respectively -- with an accelerated mirror-prox algorithm as the solver in each inner loop. We establish a global convergence guarantee for solving monotone and strongly monotone NGNEPs and provide nonasymptotic complexity bounds expressed in terms of the number of gradient evaluations. Experimental results demonstrate the efficiency of our algorithms in practice.

翻译：我们研究一类非线性广义纳什均衡问题（NGNEPs）中均衡的计算问题，其中每个参与者的策略集由可能依赖于竞争对手选择的等式和不等式约束定义。尽管解决该问题的算法的渐近全局收敛性和局部收敛率已得到广泛研究，但非渐近迭代复杂度的分析仍处于起步阶段。本文提出了两种一阶算法——分别基于二次罚函数法（QPM）和增广拉格朗日法（ALM）——并在每个内循环中使用加速镜像投影算法作为求解器。我们建立了求解单调和强单调NGNEPs的全局收敛性保证，并给出了以梯度评估次数表示的非渐近复杂度界。实验结果表明了我们的算法在实际中的高效性。