In this study, we investigate the performance of the Metropolis-adjusted Langevin algorithm in a setting with constraints on the support of the target distribution. We provide a rigorous analysis of the resulting Markov chain, establishing its convergence and deriving an upper bound for its mixing time. Our results demonstrate that the Metropolis-adjusted Langevin algorithm is highly effective in handling this challenging situation: the mixing time bound we obtain is superior to the best known bounds for competing algorithms without an accept-reject step. Our numerical experiments support these theoretical findings, indicating that the Metropolis-adjusted Langevin algorithm shows promising performance when dealing with constraints on the support of the target distribution.

翻译：在本研究中，我们探讨了Metropolis-adjusted Langevin算法在目标分布支撑集存在约束条件下的性能表现。我们对由此生成的马尔可夫链进行了严谨分析，验证了其收敛性并推导出混合时间的上界。研究结果表明，Metropolis-adjusted Langevin算法在处理此类具有挑战性的情形时表现出色：我们获得的混合时间上界优于已知最优的（不含接受-拒绝步骤的）竞争算法上界。数值实验进一步验证了这些理论发现，表明该算法在处理目标分布支撑集的约束条件时展现出优越性能。