We characterize the convergence properties of traditional best-response (BR) algorithms in computing solutions to mixed-integer Nash equilibrium problems (MI-NEPs) that turn into a class of monotone Nash equilibrium problems (NEPs) once relaxed the integer restrictions. We show that the sequence produced by a Jacobi/Gauss-Seidel BR method always approaches a bounded region containing the entire solution set of the MI-NEP, whose tightness depends on the problem data, and it is related to the degree of strong monotonicity of the relaxed NEP. When the underlying algorithm is applied to the relaxed NEP, we establish data-dependent complexity results characterizing its convergence to the unique solution of the NEP. In addition, we derive one of the very few sufficient conditions for the existence of solutions to MI-NEPs. The theoretical results developed bring important practical advantages that are illustrated on a numerical instance of a smart building control application.

翻译：我们刻画了传统最优响应（BR）算法在求解混合整数纳什均衡问题（MI-NEPs）时的收敛性质，此类问题在松弛整数约束后转化为一类单调纳什均衡问题（NEPs）。我们证明：Jacobi/Gauss-Seidel BR方法产生的序列始终趋近于包含MI-NEP整个解集的有界区域，该区域的紧致性取决于问题数据，并与松弛NEP的强单调性程度相关。当底层算法应用于松弛NEP时，我们建立了依赖数据的复杂度结果，刻画其收敛至NEP唯一解的特性。此外，我们推导出了MI-NEP解存在的极少数充分条件之一。所发展的理论成果带来了重要的实践优势，并在智能建筑控制应用的数值实例中得到了验证。