# Convex Functions and Optimization Methods on Riemannian Manifolds

• Constantin Udrişte Book

Part of the Mathematics and Its Applications book series (MAIA, volume 297)

1. Front Matter
Pages i-xvii
2. Constantin Udrişte
Pages 1-33
3. Constantin Udrişte
Pages 34-55
4. Constantin Udrişte
Pages 56-107
5. Constantin Udrişte
Pages 108-127
6. Constantin Udrişte
Pages 128-185
7. Constantin Udrişte
Pages 186-225
8. Constantin Udrişte
Pages 226-286
9. Back Matter
Pages 287-350

### Introduction

The object of this book is to present the basic facts of convex functions, standard dynamical systems, descent numerical algorithms and some computer programs on Riemannian manifolds in a form suitable for applied mathematicians, scientists and engineers. It contains mathematical information on these subjects and applications distributed in seven chapters whose topics are close to my own areas of research: Metric properties of Riemannian manifolds, First and second variations of the p-energy of a curve; Convex functions on Riemannian manifolds; Geometric examples of convex functions; Flows, convexity and energies; Semidefinite Hessians and applications; Minimization of functions on Riemannian manifolds. All the numerical algorithms, computer programs and the appendices (Riemannian convexity of functions f:R ~ R, Descent methods on the Poincare plane, Descent methods on the sphere, Completeness and convexity on Finsler manifolds) constitute an attempt to make accesible to all users of this book some basic computational techniques and implementation of geometric structures. To further aid the readers,this book also contains a part of the folklore about Riemannian geometry, convex functions and dynamical systems because it is unfortunately "nowhere" to be found in the same context; existing textbooks on convex functions on Euclidean spaces or on dynamical systems do not mention what happens in Riemannian geometry, while the papers dealing with Riemannian manifolds usually avoid discussing elementary facts. Usually a convex function on a Riemannian manifold is a real­ valued function whose restriction to every geodesic arc is convex.

### Keywords

Convexity Mathematica Riemannian geometry algorithm algorithms differential equation dynamical systems dynamische Systeme geometry manifold modeling numerical analysis optimization programming

#### Authors and affiliations

• Constantin Udrişte
• 1
1. 1.Department of MathematicsPolitehnica University of BucharestBucharestRomania

### Bibliographic information

• DOI https://doi.org/10.1007/978-94-015-8390-9