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 benefits, illustrated on a numerical instance of a smart building control application.

翻译：本文研究了传统最佳响应算法在求解混合整数纳什均衡问题时的收敛特性，这类问题在松弛整数约束后会转化为一类单调纳什均衡问题。我们证明，雅可比/高斯-赛德尔最佳响应算法生成的序列始终趋近于包含混合整数纳什均衡问题全部解集的有界区域，该区域的紧致度取决于问题数据，且与松弛后纳什均衡问题的强单调性程度相关。当基础算法应用于松弛后的纳什均衡问题时，我们建立了数据依赖的复杂度结果以刻画其收敛到纳什均衡问题唯一解的特性。此外，我们推导出混合整数纳什均衡问题解存在的充分条件（此类条件在现有文献中极为罕见）。所发展的理论结果具有重要实用价值，本文通过智能建筑控制应用的数值算例对此进行了说明。