SIAM Journal on Control and Optimization, Volume 62, Issue 2, Page 1093-1121, April 2024.
Abstract. In this paper, we propose consensus-based optimization for saddle point problems (CBO-SP), a novel multi-particle metaheuristic derivative-free optimization method capable of provably finding global Nash equilibria. Following the idea of swarm intelligence, the method employs two groups of interacting particles, one which performs a minimization over one variable while the other performs a maximization over the other variable. The two groups constantly exchange information through a suitably weighted average. This paradigm permits a passage to the mean-field limit, which makes the method amenable to theoretical analysis, and it allows to obtain rigorous convergence guarantees under reasonable assumptions about the initialization and the objective function, which most notably include nonconvex-nonconcave objectives. We further provide numerical evidence for the success of the algorithm.

中文翻译：

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基于共识的鞍点问题优化

SIAM 控制与优化杂志，第 62 卷，第 2 期，第 1093-1121 页，2024 年 4 月。
摘要。在本文中，我们提出了基于共识的鞍点问题优化（CBO-SP），这是一种新颖的多粒子元启发式无导数优化方法，能够证明找到全局纳什均衡。遵循群体智能的思想，该方法采用两组相互作用的粒子，一组对一个变量执行最小化，而另一组对另一个变量执行最大化。两组通过适当的加权平均值不断交换信息。这种范式允许达到平均场极限，这使得该方法适合理论分析，并且它允许在关于初始化和目标函数的合理假设下获得严格的收敛保证，其中最显着的是包括非凸非凹目标。我们进一步为该算法的成功提供了数字证据。