We present convergence and error estimates of modified versions of the time-discrete consensus-based optimization (CBO) algorithm proposed in Carrillo et al. (ESAIM: Control Optim Calc Var, 2020) for general non-convex functions. In authors’ recent work (Ha et al. in Math Models Meth Appl Sci 30:2417–2444, 2020), rigorous error analysis of a modified version of the first-order consensus-based optimization algorithm proposed in Carrillo et al. (2020) was studied at the particle level without resorting to the kinetic equation via a mean-field limit. However, the error analysis for the corresponding time- discrete algorithm was not done mainly due to lack of discrete analogue of Itô’s stochastic calculus. In this paper, we provide a simple and elementary convergence and error analysis for a general time-discrete consensus-based optimization algorithm, which includes modifications of the three discrete algorithms in Carrillo et al. (2020), two of which are present in Ha et al. (2020). Our analysis provides numerical stability and convergence conditions for the three algorithms, as well as error estimates to the global minimum.

中文翻译：

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基于时间离散共识的优化算法的收敛和误差估计

我们提出了 Carrillo 等人提出的基于时间离散共识的优化 (CBO) 算法的修改版本的收敛和误差估计。（ESAIM：Control Optim Calc Var，2020）用于一般非凸函数。在作者最近的工作中（Ha et al. in Math Models Meth Appl Sci 30:2417–2444, 2020），对 Carrillo 等人提出的一阶共识优化算法的修改版本进行了严格的误差分析。(2020) 在粒子水平上进行了研究，而没有通过平均场限制求助于动力学方程。然而，由于缺乏伊藤随机演算的离散模拟，因此没有对相应的时间离散算法进行误差分析。在本文中，我们为通用的基于时间离散一致性的优化算法提供了一个简单而基本的收敛和误差分析，其中包括对 Carrillo 等人的三种离散算法的修改。（2020），其中两个存在于 Ha 等人中。(2020)。我们的分析为三种算法提供了数值稳定性和收敛条件，以及对全局最小值的误差估计。