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.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.Keywordssaddle point problemsNash equilibria nonconvex-nonconcavederivative-free optimizationmetaheuristicsconsensus-based optimizationFokker–Planck equationsMSC codes90C4765C3565K0590C5635Q9035Q83
Huang et al. (Mon,) studied this question.