Convex Functions and Optimization Methods on Riemannian Manifolds

1994
ISBN: 978-0-7923-3002-8
Verlag: Springer Netherlands

Buch, Englisch, Band 297, 350 Seiten, HC runder Rücken kaschiert, Format (B × H): 160 mm x 241 mm, Gewicht: 1540 g

Reihe: Mathematics and Its Applications

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.

Zielgruppe

Research

Autoren/Hrsg.

Udriste, C.

Fachgebiete

• Mathematik | Informatik EDV | Informatik Informatik
• Mathematik | Informatik Mathematik Geometrie Differentialgeometrie
• Mathematik | Informatik Mathematik Mathematische Analysis Elementare Analysis und Allgemeine Begriffe
• Mathematik | Informatik Mathematik Mathematische Analysis Variationsrechnung
• Mathematik | Informatik Mathematik Operations Research Spieltheorie
• Technische Wissenschaften Maschinenbau | Werkstoffkunde Produktionstechnik Fertigungstechnik
• Mathematik | Informatik Mathematik Geometrie Elementare Geometrie: Allgemeines
• Technische Wissenschaften Technik Allgemein Mathematik für Ingenieure

Inhaltsverzeichnis

1. Metric properties of Riemannian manifolds.- 2. First and second variations of the p-energy of a curve.- 3. Convex functions on Riemannian manifolds.- 4. Geometric examples of convex functions.- 5. Flows, convexity and energies.- 6. Semidefinite Hessians and applications.- 7. Minimization of functions on Riemannian manifolds.- Appendices.- 1. Riemannian convexity of functions f: ? ? ?.- §0. Introduction.- §1. Geodesics of (?, g).- §3. Convex functions on (?, g).- 2. Descent methods on the Poincaré plane.- §0. Introduction.- §1. Poincaré plane.- §2. Linear affine functions on the Poincaré plane.- §3. Quadratic affine functions on the Poincaré plane.- §4. Convex functions on the Poincaré plane.- Examples of hyperbolic convex functions.- §5. Descent algorithm on the Poincaré plane.- TC program for descent algorithm on Poincaré plane (I).- TC program f or descent algorithm on Poincaré plane (II).- 3. Descent methods on the sphere.- §1. Gradient and Hessian on the sphere.- §2. Descent algorithm on the sphere.- Critical values of the normal stress.- Critical values of the shear stress.- TC program for descent method on the unit sphere.- 4. Completeness and convexity on Finsler manifolds.- §1. Complete Finsler manifolds.- §2. Analytical criterion for completeness.- §3. Warped products of complete Finsler manifolds.- §4. Convex functions on Finsler manifolds.- References.