Second Order Necessary Conditions for Endpoints-Constrained Optimal Control Problems on Riemannian manifolds

Efficiency for multitime vector variational problems on Riemannian manifolds involving geodesic quasiinvex functionals

We study the connection between a multitime scalar variational problem (SVP), a multitime vector variational problem (VVP) and a multitime vector fractional variational problem (VFP). For (SVP), we establish necessary optimality conditions. For both vector variational problems, we define the notions of Pareto efficient solution and of normal efficient solution and we establish necessary efficiency conditions for (VVP) and (VFP) using both notions. The main purpose of the paper is to establish sufficient efficiency conditions for the vector problems (VVP) and (VFP). Moreover, we obtain sufficie…

Minimum $L^\infty$ Accelerations in Riemannian Manifolds

Riemannian cubics are critical points for the $L^2$ norm of acceleration of curves in Riemannian manifolds $M$. In the present paper the $L^\infty$ norm replaces the $L^2$ norm, and a less direct argument is used to derive necessary conditions analogous to those for Riemannian cubics. The necessary conditions are examined when $M$ is a sphere or a bi-invariant Lie group.

Backward variational approach on time scales with an action depending on the free endpoints

We establish necessary optimality conditions for variational problems with an action depending on the free endpoints. New transversality conditions are also obtained. The results are formulated and proved using the recent and general theory of time scales via the backward nabla differential operator.

黎曼流形上区间值函数的广义Hukuhara方向可微性

Generalized Hukuhara directional differentiability of interval-valued functions on Riemannian manifolds

该研究指出黎曼流形上区间值函数的广义Hukuhara方向可微性不等价于其端点、中心及半宽函数的方向可微性，纠正了相关文献的论断，并给出区间值函数凸性与方向可微性的基础结果。

Local Minimizers for Variational Obstacle Avoidance on Riemannian manifolds

This paper studies a variational obstacle avoidance problem on complete Riemannian manifolds. That is, we minimize an action functional, among a set of admissible curves, which depends on an artificial potential function used to avoid obstacles. In particular, we generalize the theory of bi-Jacobi fields and biconjugate points and present necessary and sufficient conditions for optimality. Local minimizers of the action functional are divided into two categories and subsequently classified, with local uniqueness results obtained in both cases.

Variational problems on Riemannian manifolds with constrained accelerations

We introduce variational problems on Riemannian manifolds with constrained acceleration and derive necessary conditions for normal extremals in the constrained variational problem. The problem consists on minimizing a higher-order energy functional, among a set of admissible curves defined by a constraint on the covariant acceleration. In addition, we use this framework to address the elastic splines problem with obstacle avoidance in the presence of this type of contraints.

Stiefel流形及其商流形上计算黎曼对数的闭式测地线与信赖域方法

Closed-form geodesics and trust-region method to calculate Riemannian logarithms on Stiefel and its quotient manifolds

本文针对Stiefel流形及其商流形提出闭式测地线公式，并结合信赖域优化平方Frobenius距离来计算黎曼对数映射与测地距离，为无闭式对数映射的流形提供了新计算框架。

Robust geometric modeling of 3-periodic tensegrity frameworks using Riemannian optimization

Framework materials and their deformations provide a compelling relation between materials science and algebraic geometry. Physical distance constraints within the material transform into polynomial constraints, making algebraic geometry and associated numerical strategies feasible for finding equilibrium configurations and deformation pathways. In this paper, we build the necessary geometric formulations and numerical strategies to explore the mechanics of two examples of 3-periodic tensegrity frameworks through non-linear optimization, eventually showing that the structures are auxetic by mu…

Reduction by Symmetry in Obstacle Avoidance Problems on Riemannian Manifolds

This paper studies the reduction by symmetry of a variational obstacle avoidance problem. We derive the reduced necessary conditions in the case of Lie groups endowed with a left-invariant metric, and for its corresponding Riemannian homogeneous spaces by considering an alternative variational problem written in terms of a connection on the horizontal bundle of the Lie group. A number of special cases where the obstacle avoidance potential can be computed explicitly are studied in detail, and these ideas are applied to the obstacle avoidance task for a rigid body evolving on SO$(3)$ and for th…

低秩矩阵补全的黎曼优化方法——扩展版

Low-rank matrix completion by Riemannian optimization---extended version

提出一种基于黎曼流形上非线性共轭梯度的矩阵补全算法，利用度量投影作为回缩，向量传输获取共轭搜索方向，并在受限等距性质下证明正则化版本的收敛性，数值实验表明算法在大规模问题上扩展性好且优于现有方法。

黎曼流形上的变分障碍回避问题

Variational obstacle avoidance problem on Riemannian manifolds

本文研究了黎曼流形上的变分障碍回避问题，推导了正常极值存在的必要条件，探讨了两种场景：一般黎曼流形和子黎曼问题，并应用于刚体和无人车的障碍回避。