A Riemannian smoothing steepest descent method for non-Lipschitz optimization on submanifolds

Continuous-Discrete Filtering and Smoothing on Submanifolds of Euclidean Space

In this paper the issue of filtering and smoothing in continuous discrete time is studied when the state variable evolves in some submanifold of Euclidean space, which may not have the usual Lebesgue measure. Formal expressions for prediction and smoothing problems are derived, which agree with the classical results except that the formal adjoint of the generator is different in general. For approximate filtering and smoothing the projection approach is taken, where it turns out that the prediction and smoothing equations are the same as in the case when the state variable evolves in Euclidean…

嵌入欧氏空间的紧黎曼子流形上非光滑优化问题的黎曼光滑化梯度型算法

Riemannian Smoothing Gradient Type Algorithms]{Riemannian Smoothing Gradient Type Algorithms for Nonsmooth Optimization Problem on Compact Riemannian Submanifold Embedded in Euclidean Space

本文针对嵌入欧氏空间的紧黎曼子流形上的非凸非光滑复合优化问题，提出基于Moreau包络的黎曼光滑化梯度型算法，并证明其迭代复杂度分别达到O(ε^{-3})和O(ε^{-5})。

黎曼子流形上降低计算成本的修正Armijo线搜索优化方法

Modified Armijo line search in optimization on Riemannian submanifolds with reduced computational cost

本文针对欧氏空间黎曼子流形上的优化问题，提出一种仅在必要时才计算收缩映射的新型黎曼线搜索策略，并证明基于该策略的最速下降法和牛顿法具有全局收敛性，数值实验表明其优于标准黎曼Armijo线搜索方法。

AdaDGS: An adaptive black-box optimization method with a nonlocal directional Gaussian smoothing gradient

The local gradient points to the direction of the steepest slope in an infinitesimal neighborhood. An optimizer guided by the local gradient is often trapped in local optima when the loss landscape is multi-modal. A directional Gaussian smoothing (DGS) approach was recently proposed in (Zhang et al., 2020) and used to define a truly nonlocal gradient, referred to as the DGS gradient, for high-dimensional black-box optimization. Promising results show that replacing the traditional local gradient with the DGS gradient can significantly improve the performance of gradient-based methods in optimi…

Optimization on the symplectic group

We regard the real symplectic group $Sp(2n,\mathbb{R})$ as a constraint submanifold of the $2n\times 2n$ real matrices $\mathcal{M}_{2n}(\mathbb{R})$ endowed with the Euclidean (Frobenius) metric, respectively as a submanifold of the general linear group $Gl(2n,\mathbb{R})$ endowed with the (left) invariant metric. For a cost function that defines an optimization problem on the real symplectic group we give a necessary and sufficient condition for critical points and we apply this condition to the particular case of a least square cost function. In order to characterize the critical points we …

用于非光滑优化的自适应黎曼 ADMM：无需平滑的最优复杂度

Adaptive Riemannian ADMM for Nonsmooth Optimization: Optimal Complexity without Smoothing

针对紧致黎曼子流形上的光滑与非光滑凸复合优化问题，提出无需平滑非光滑项的自适应 ARADMM，每次迭代仅需一次梯度求值和一次近端更新，并达到 O(ε^-3) 的最优 KKT 复杂度。