研究广义外梯度法在极小极大问题中的不动点性质，揭示其稳定解与鞍点的关系。
Properties of Fixed Points of Generalised Extra Gradient Methods Applied to Min-Max Problems
- 通过稳定性分析，建立广义外梯度法的不动点与鞍点的关联
- 在合适步长下，鞍点构成算法稳定不动点的子集
- 适用于博弈论、优化等需求解鞍点的场景
本文研究了广义外梯度（GEG）算法应用于极小极大问题时的不动点性质。探讨了极小极大问题目标函数的鞍点与GEG不动点之间的联系。结果表明，在合适的步长选择下，鞍点（纳什均衡）集合是GEG稳定不动点的子集。通过离散时间动力系统的稳定性分析，获得了GEG算法的收敛性性质。数值实验展示了该方法相较于现有方法的优势与有效性。
原文摘要 · Abstract (English)
This paper studies properties of fixed points of generalised Extra-gradient (GEG) algorithms applied to min-max problems. We discuss connections between saddle points of the objective function of the min-max problem and GEG fixed points. We show that, under appropriate step-size selections, the set of saddle points (Nash equilibria) is a subset of stable fixed points of GEG. Convergence properties of the GEG algorithm are obtained through a stability analysis of a discrete-time dynamical system. The results and benefits when compared to existing methods are illustrated through numerical examples.
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