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 of the GNEP, we construct a set of convexified instances and show that a feasible strategy profile is an equilibrium for the original instance if and only if it is an equilibrium for any convexified instance and the convexified cost functions coincide with the initial ones. We develop this convexification approach along three dimensions: We first show that for quasi-linear models, where a convexified instance exists in which for fixed strategies of the opponent players, the cost function of every player is linear and the respective strategy space is polyhedral, the convexification reduces the GNEP to a standard (non-linear) optimization problem. Secondly, we derive two complete characterizations of those GNEPs for which the convexification leads to a jointly constrained or a jointly convex GNEP, respectively. These characterizations require new concepts related to the interplay of the convex hull operator applied to restricted subsets of feasible strategies and may be interesting on their own. Note that this characterization is also computationally relevant as jointly convex GNEPs have been extensively studied in the literature. Finally, we demonstrate the applicability of our results by presenting a numerical study regarding the computation of equilibria for three classes of GNEPs related to integral network flows and discrete market equilibria.

翻译：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）。这类广义博弈包含重要的混合整数变量博弈情形，而关于此类博弈的文献中仅存在少量已知结论。我们提出一种新方法，通过利用Nikaido-Isoda函数进行凸化处理来刻画均衡特征。对于任意给定的GNEP实例，我们构造一组凸化实例，并证明：一个可行策略组合是原始实例的均衡当且仅当其同时是任意凸化实例的均衡，且凸化后的成本函数与初始函数完全一致。我们从三个维度展开该凸化方法：首先证明，对于拟线性模型——即存在一个凸化实例使得：当对手策略固定时，每个玩家的成本函数均为线性且对应策略空间为多面体——凸化将GNEP转化为标准（非线性）优化问题；其次，我们分别给出两类GNEP的完整刻画条件，这两类GNEP的凸化过程将分别产生联合约束型或联合凸型GNEP。这些刻画需要引入关于可行策略受限子集凸包算子相互作用的新概念，其本身可能具有理论价值。值得注意的是，由于联合凸型GNEP在文献中已被广泛研究，该刻画条件同样具有计算相关性；最后，通过三类与积分网络流和离散市场均衡相关的GNEP均衡计算数值研究，我们验证了所得结果的适用性。