Convex Functions and Optimization Methods on Riemannian Manifolds

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.