We introduce a new stochastic differential model for global optimization of nonconvex functions on compact hypersurfaces. The model is inspired by the stochastic Kuramoto–Vicsek system and belongs to the class of Consensus-Based Optimization methods. In fact, particles move on the hypersurface driven by a drift towards an instantaneous consensus point, computed as a convex combination of the particle locations weighted by the cost function according to Laplace’s principle. The consensus point represents an approximation to a global minimizer. The dynamics is further perturbed by a random vector field to favor exploration, whose variance is a function of the distance of the particles to the consensus point. In particular, as soon as the consensus is reached, then the stochastic component vanishes. In this paper, we study the well-posedness of the model and we derive rigorously its mean-field approximation for large particle limit.

中文翻译：

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基于共识的超曲面优化：适定性和平均场极限

我们引入了一种新的随机微分模型，用于在紧凑超曲面上对非凸函数进行全局优化。该模型受到随机 Kuramoto-Vicsek 系统的启发，属于基于共识的优化方法类。事实上，粒子在超曲面上移动是由向瞬时一致性点的漂移驱动的，根据拉普拉斯原理，计算为由成本函数加权的粒子位置的凸组合。共识点表示对全局最小化器的近似。动力学进一步受到随机向量场的扰动以利于探索，其方差是粒子到共识点的距离的函数。特别是，一旦达成共识，随机分量就会消失。在本文中，