We study Nash equilibrium problems with mixed-integer variables in which each player solves a mixed-integer optimization problem parameterized by the rivals' strategies. We distinguish between standard Nash equilibrium problems (NEPs), where parameterization affects only the objective functions, and generalized Nash equilibrium problems (GNEPs), where strategy sets may additionally depend on rivals' strategies. We introduce a branch-and-cut (B&C) algorithm for such mixed-integer games that, upon termination, either computes a pure Nash equilibrium or decides their non-existence. Our approach reformulates the game as a bilevel problem using the Nikaido--Isoda function. We then use bilevel-optimization techniques to get a computationally tractable relaxation of this reformulation and embed it into a B&C framework. We derive sufficient conditions for the existence of suitable cuts and finite termination of our method depending on the setting. For GNEPs, we adapt the idea of intersection cuts from bilevel optimization and mixed-integer linear optimization. We can guarantee the existence of such cuts under suitable assumptions, which are particularly fulfilled for pure-integer GNEPs with decoupled concave objectives and linear coupling constraints. For NEPs, we show that suitable cuts always exist via best-response inequalities and prove that our B&C method terminates in finite time whenever the set of best-response sets is finite. We show that this condition is fulfilled for the important special cases of (i) players' cost functions being concave in their own continuous strategies and (ii) the players' cost functions only depending on their own strategy and the rivals' integer strategy components. Finally, we present preliminary numerical results for two different types of knapsack games, a game based on capacitated flow problems, and integer NEPs with quadratic objectives.

翻译：本文研究具有混合整数变量的纳什均衡问题，其中每个参与者需解决一个以对手策略为参数的混合整数优化问题。我们区分标准纳什均衡问题（NEPs，其参数化仅影响目标函数）与广义纳什均衡问题（GNEPs，其策略集可能额外依赖于对手策略）。针对此类混合整数博弈，我们提出一种分支割平面算法，该算法在终止时要么计算出一个纯纳什均衡，要么判定其不存在性。我们的方法通过Nikaido-Isoda函数将博弈重构为双层规划问题，继而运用双层优化技术获得该重构问题的计算易处理松弛形式，并将其嵌入分支割平面框架。我们推导出算法存在适用割平面及有限终止的充分条件，这些条件取决于具体问题设定。对于广义纳什均衡问题，我们借鉴双层优化与混合整数线性规划中的交割思想，在适当假设下可保证此类割平面的存在性，该假设在具有解耦凹目标函数与线性耦合约束的纯整数广义纳什均衡问题中尤其成立。对于标准纳什均衡问题，我们通过最优响应不等式证明适用割平面始终存在，并证明当最优响应集集合为有限集时，我们的分支割平面算法必在有限时间内终止。我们进一步证明该条件在以下重要特例中成立：（i）参与者成本函数关于自身连续策略呈凹性；（ii）参与者成本函数仅取决于自身策略及对手的整数策略分量。最后，我们针对两类背包博弈、基于容量约束流问题的博弈以及具有二次目标的整数纳什均衡问题，给出了初步数值计算结果。