We consider generalized Nash equilibrium problems (GNEPs) with non-convex strategy spaces and non-convex cost functions. This general class of games includes the important case of games with mixed-integer variables for which only a few results are known in the literature. We present a new approach to characterize equilibria via a convexification technique using the Nikaido-Isoda function. To any given instance $I$ of the GNEP, we derive a convexified instance $I^\text{conv}$ and show that every feasible strategy profile for $I$ is an equilibrium if and only if it is an equilibrium for $I^\text{conv}$ and the convexified cost functions coincide with the initial ones. Based on this general result we identify important classes of GNEPs which allow us to reformulate the equilibrium problem via standard optimization problems. $1.$ First, quasi-linear GNEPs are introduced where for fixed strategies of the opponent players, the cost function of every player is linear and the convex hull of the respective strategy space is polyhedral. For this game class we reformulate the equilibrium problem for $I^\text{conv}$ as a standard (non-linear) optimization problem. $2.$ Secondly, we study GNEPs with joint constraint sets. We introduce the new class of projective-closed GNEPs for which we show that $I^\text{conv}$ falls into the class of jointly convex GNEPs. As an important application, we show that general GNEPs with shared binary sets $\{0,1\}^k$ are projective-closed. $3.$ Thirdly, we discuss the class of quasi-separable GNEPs in which roughly speaking the players' cost functions depend on their own strategy only. We show that they admit a special structure leading to a characterization of equilibria via solutions of a convex optimization problem. $4.$ Finally, we present numerical results regarding the computation of equilibria for a class of quasi-linear and projective-closed GNEPs.

中文翻译：

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混合整数变量的广义纳什均衡问题

我们考虑具有非凸策略空间和非凸成本函数的广义纳什均衡问题 (GNEP)。这类一般博弈包括具有混合整数变量的博弈的重要情况，在文献中只有少数结果是已知的。我们提出了一种通过使用 Nikaido-Isoda 函数的凸化技术来表征平衡的新方法。对于 GNEP 的任何给定实例 $I$，我们导出一个凸化实例 $I^\text{conv}$ 并表明 $I$ 的每个可行策略配置文件都是均衡当且仅当它是 $I 的均衡^\text{conv}$ 和凸成本函数与初始值一致。基于这个一般结果，我们确定了重要的 GNEP 类别，这些类别使我们能够通过标准优化问题重新制定平衡问题。$1.$ 首先，引入了准线性 GNEP，其中对于对手玩家的固定策略，每个玩家的成本函数是线性的，各自策略空间的凸包是多面体。对于这个游戏类，我们将 $I^\text{conv}$ 的均衡问题重新表述为标准（非线性）优化问题。$2.$ 其次，我们研究具有联合约束集的 GNEP。我们介绍了一类新的射影封闭 GNEP，我们证明 $I^\text{conv}$ 属于联合凸 GNEP 类。作为一个重要的应用，我们展示了具有共享二进制集 $\{0,1\}^k$ 的一般 GNEP 是射影封闭的。$3.$ 第三，我们讨论准可分离GNEPs 类，其中粗略地说，参与者的成本函数仅取决于他们自己的策略。我们表明，他们承认一个特殊的结构，导致通过凸优化问题的解决方案来表征均衡。$4.$ 最后，我们给出了关于一类准线性和射影封闭 GNEP 的平衡计算的数值结果。