Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the supporting quasi-hyperplane and separation theorem of geodesic convex sets with applications on Riemannian manifolds

Li-wen Zhou, Ling-ling Liu, Chao Min, Yao-jia Zhang, Nan-Jing Huang

Abstract

In this paper, we first establish the separation theorem between a point and a locally geodesic convex set and then prove the existence of a supporting quasi-hyperplane at any point on the boundary of the closed locally geodesic convex set on a Riemannian manifold. As applications, some optimality conditions are obtained for optimization problems with constraints on Riemannian manifolds.

On the supporting quasi-hyperplane and separation theorem of geodesic convex sets with applications on Riemannian manifolds

Abstract

In this paper, we first establish the separation theorem between a point and a locally geodesic convex set and then prove the existence of a supporting quasi-hyperplane at any point on the boundary of the closed locally geodesic convex set on a Riemannian manifold. As applications, some optimality conditions are obtained for optimization problems with constraints on Riemannian manifolds.
Paper Structure (7 sections, 17 theorems, 50 equations)

This paper contains 7 sections, 17 theorems, 50 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Separation theorem on $M$
  4. Applications
  5. Tangent Cone and Normal Cone
  6. Optimality conditions of optimization problems with constraints
  7. Conclusions

Key Result

Lemma 2.1

(llm2, Lemma 2.4) Let $x_0\in M$ and $\{x_n\}\subset M$ such that $x_n\rightarrow x_0$. Then, the following assertions hold.

Theorems & Definitions (40)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.1
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Remark 2.1
  • Definition 2.7
  • Lemma 2.2
  • ...and 30 more