In this paper, we consider stochastic monotone Nash games where each player’s strategy set is characterized by possibly a large number of explicit convex constraint inequalities. Notably, the functional constraints of each player may depend on the strategies of other players, allowing for capturing a subclass of generalized Nash equilibrium problems (GNEP). While there is limited work that provide guarantees for this class of stochastic GNEPs, even when the functional constraints of the players are independent of each other, the majority of the existing methods rely on employing projected stochastic approximation (SA) methods. However, the projected SA methods perform poorly when the constraint set is afflicted by the presence of a large number of possibly nonlinear functional inequalities. Motivated by the absence of performance guarantees for computing the Nash equilibrium in constrained stochastic monotone Nash games, we develop a single timescale randomized Lagrangian multiplier stochastic approximation method where in the primal space, we employ an SA scheme, and in the dual space, we employ a randomized block-coordinate scheme where only a randomly selected Lagrangian multiplier is updated. We show that our method achieves a convergence rate of \(\mathcal {O}\left( \frac{\log (k)}{\sqrt{k}}\right)\) for suitably defined suboptimality and infeasibility metrics in a mean sense.

在本文中，我们考虑随机单调纳什博弈，其中每个玩家的策略集可能具有大量显式凸约束不等式。值得注意的是，每个参与者的功能约束可能取决于其他参与者的策略，从而允许捕获广义纳什均衡问题（GNEP）的子类。尽管为此类随机 GNEP 提供保证的工作有限，但即使参与者的功能约束彼此独立，大多数现有方法仍依赖于采用投影随机近似（SA）方法。然而，当约束集受到大量可能非线性函数不等式的影响时，投影 SA 方法表现不佳。由于缺乏在约束随机单调纳什博弈中计算纳什均衡的性能保证，我们开发了一种单时间尺度随机拉格朗日乘子随机逼近方法，其中在原始空间中，我们采用 SA 方案，在对偶空间中，我们采用随机块坐标方案，其中仅更新随机选择的拉格朗日乘数。我们表明，对于适当定义的次优性和不可行性指标，我们的方法达到了\(\mathcal {O}\left( \frac{\log (k)}{\sqrt{k}}\right)\)的收敛速度意思是意思。