Convex Functions and Optimization Methods on Riemannian Manifolds

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.

Klappentext

This unique monograph discusses the interaction between Riemannian geometry, convex programming, numerical analysis, dynamical systems and mathematical modelling. The book is the first account of the development of this subject as it emerged at the beginning of the 'seventies. A unified theory of convexity of functions, dynamical systems and optimization methods on Riemannian manifolds is also presented. Topics covered include geodesics and completeness of Riemannian manifolds, variations of the p-energy of a curve and Jacobi fields, convex programs on Riemannian manifolds, geometrical constructions of convex functions, flows and energies, applications of convexity, descent algorithms on Riemannian manifolds, TC and TP programs for calculations and plots, all allowing the user to explore and experiment interactively with real life problems in the language of Riemannian geometry. An appendix is devoted to convexity and completeness in Finsler manifolds. For students and researchers in such diverse fields as pure and applied mathematics, and applied sciences like physics, chemistry, biology, and engineering.

Inhalt

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.