标题： 对随机非凸约束问题的内部镜像下降流的探索 显示英文标题

标题： On exploration of an interior mirror descent flow for stochastic nonconvex constrained problem

摘要： 我们研究一个在非凸约束上定义的非光滑非凸优化问题，其中可行集由一个开集的闭包与一个光滑流形的交集给出。 通过赋予开集一个由障碍函数诱导的黎曼度量，我们得到一个作为微分包含形式的黎曼次梯度流，该流始终保持在可行集的内部。 这个连续的动力系统通过揭示这些方法可以被解释为连续流的离散近似，从而统一了两类迭代优化方法，即Hessian障碍法和镜像下降方案。 我们探讨由这个动力系统生成的轨迹的长期行为，并表明Hessian障碍法和镜像下降方案现有的收敛性质不足可以通过连续轨迹的特性得到统一且更深入的解释。 例如，在Hessian障碍法和镜像下降方案中观察到的著名虚假平稳点\cite{chen2024spurious}被解释为动力系统的稳定平衡点，这些平衡点并不对应于原始优化问题的实际平稳点。 我们提供了两个充分条件，使得如果满足严格互补条件，就可以避免这些虚假平稳点。 在缺乏这些正则性条件的情况下，我们提出了一种随机扰动策略，以确保轨迹收敛（子序列地）到一个近似平稳点。 基于这些见解，我们引入了两种迭代的黎曼次梯度方法，属于内点法的形式，这些方法推广了现有的Hessian障碍法和镜像下降方案，用于求解非光滑非凸优化问题。 显示英文摘要

摘要： We study a nonsmooth nonconvex optimization problem defined over nonconvex constraints, where the feasible set is given by the intersection of the closure of an open set and a smooth manifold. By endowing the open set with a Riemannian metric induced by a barrier function, we obtain a Riemannian subgradient flow formulated as a differential inclusion, which remains strictly within the interior of the feasible set. This continuous dynamical system unifies two classes of iterative optimization methods, namely the Hessian barrier method and mirror descent scheme, by revealing that these methods can be interpreted as discrete approximations of the continuous flow. We explore the long-term behavior of the trajectories generated by this dynamical system and show that the existing deficient convergence properties of the Hessian barrier and mirror descent scheme can be unifily and more insightfully interpreted through these of the continuous trajectory. For instance, the notorious spurious stationary points \cite{chen2024spurious} observed in Hessian barrier method and mirror descent scheme are interpreted as stable equilibria of the dynamical system that do not correspond to real stationary points of the original optimization problem. We provide two sufficient condition such that these spurious stationary points can be avoided if the strict complementarity conditions holds. In the absence of these regularity condition, we propose a random perturbation strategy that ensures the trajectory converges (subsequentially) to an approximate stationary point. Building on these insights, we introduce two iterative Riemannian subgradient methods, form of interior point methods, that generalizes the existing Hessian barrier method and mirror descent scheme for solving nonsmooth nonconvex optimization problems.