Gradient Langevin dynamics and a variety of its variants have attracted increasing attention owing to their convergence towards the global optimal solution, initially in the unconstrained convex framework while recently even in convex constrained non-convex problems. In the present work, we extend those frameworks to non-convex problems on a non-convex feasible region with a global optimization algorithm built upon reflected gradient Langevin dynamics and derive its convergence rates. By effectively making use of its reflection at the boundary in combination with the probabilistic representation for the Poisson equation with the Neumann boundary condition, we present promising convergence rates, particularly faster than the existing one for convex constrained non-convex problems.

翻译：梯度朗之万动力学及其多种变体因收敛到全局最优解而日益受到关注，最初在无约束凸框架中，近期甚至在凸约束非凸问题中亦如此。本文中，我们基于反射梯度朗之万动力学构建的全局优化算法，将这些框架扩展到非凸可行域上的非凸问题，并推导其收敛速率。通过有效利用边界反射与带诺伊曼边界条件的泊松方程的概率表示相结合，我们提出了有前景的收敛速率，特别地，该速率快于凸约束非凸问题的现有结果。