Probably one of the most striking examples of the close connections between global optimization processes and statistical physics is the simulated annealing method, inspired by the famous Monte Carlo algorithm devised by Metropolis et al. in the middle of the last century. In this paper we show how the tools of linear kinetic theory allow to describe this gradient-free algorithm from the perspective of statistical physics and how convergence to the global minimum can be related to classical entropy inequalities. This analysis highlight the strong link between linear Boltzmann equations and stochastic optimization methods governed by Markov processes. Thanks to this formalism we can establish the connections between the simulated annealing process and the corresponding mean-field Langevin dynamics characterized by a stochastic gradient descent approach. Generalizations to other selection strategies in simulated annealing that avoid the acceptance-rejection dynamic are also provided.

翻译：全局优化过程与统计物理之间紧密联系的最显著例子之一或许是模拟退火方法，其灵感源自上世纪中叶Metropolis等人提出的著名蒙特卡罗算法。本文展示了线性动理学理论工具如何从统计物理视角描述这种无梯度算法，并揭示收敛到全局最小值与经典熵不等式之间的关联。该分析凸显了线性玻尔兹曼方程与由马尔可夫过程控制的随机优化方法之间的强相关性。借助这一形式体系，我们建立了模拟退火过程与以随机梯度下降方法为特征的相应平均场朗之万动力学之间的联系。此外，还给出了模拟退火中避免接受-拒绝动态的其他选择策略的推广形式。