Convex Functions and Optimization Methods on Riemannian Manifolds: (297 Mathematics and Its Applications)

This monograph discusses the interaction between Riemannian geometry, convex programming, numerical analysis, dynamical systems and mathematical modelling. The book is an account of the development of this subject as it emerged at the beginning of the 1970s. 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. These allow the reader 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. This text should be of interest to students and researchers in such diverse fields as pure and applied mathematics, and applied sciences like physics, chemistry, biology and engineering.

Table of Contents:
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.