Convex Functions and Optimization Methods on Riemannian Manifolds / Edition 1

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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.

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Product Details

  • ISBN-13: 9780792330028
  • Publisher: Springer Netherlands
  • Publication date: 7/31/1994
  • Series: Mathematics and Its Applications (closed) Series , #297
  • Edition description: 1994
  • Edition number: 1
  • Pages: 350
  • Product dimensions: 0.94 (w) x 6.14 (h) x 9.21 (d)

Table of Contents

Preface. 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:R—>R. 2. Descent methods on the Poincaré plane. 3. Descent methods on the sphere. 4. Completeness and convexity on Finsler manifolds. Bibliography. Index.

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