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11 - Geodesic convexity

Published online by Cambridge University Press:  09 March 2023

Nicolas Boumal
Affiliation:
École Polytechnique Fédérale de Lausanne
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Summary

Convexity is one of the most fruitful concepts in classical optimization. Geodesic convexity generalizes that concept to optimization on Riemannian manifolds. There are several ways to carry out such a generalization: This chapter favors permissive definitions which are sufficient to retain the most important properties for optimization purposes (e.g., local optima are global optima). Alternative definitions are discussed, highlighting the fact that all coincide for the special case of Hadamard manifolds (essentially, negatively curved Riemannian manifolds). The chapter continues with a discussion of the special properties of differentiable geodesically (strictly, strongly) convex functions, and builds on them to show global linear convergence of Riemannian gradient descent, assuming strong geodesic convexity and Lipschitz continuous gradients (via the Polyak–Łojasiewicz inequality). The chapter closes with two examples of manifolds where geodesic convexity has proved useful, namely, the positive orthant with a log-barrier metric (recovering geometric programming), and the cone of positive definite matrices with the log-Euclidean and the affine invariant Riemannian metrics.

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Publisher: Cambridge University Press
Print publication year: 2023

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  • Geodesic convexity
  • Nicolas Boumal, École Polytechnique Fédérale de Lausanne
  • Book: An Introduction to Optimization on Smooth Manifolds
  • Online publication: 09 March 2023
  • Chapter DOI: https://doi.org/10.1017/9781009166164.012
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  • Geodesic convexity
  • Nicolas Boumal, École Polytechnique Fédérale de Lausanne
  • Book: An Introduction to Optimization on Smooth Manifolds
  • Online publication: 09 March 2023
  • Chapter DOI: https://doi.org/10.1017/9781009166164.012
Available formats
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Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

  • Geodesic convexity
  • Nicolas Boumal, École Polytechnique Fédérale de Lausanne
  • Book: An Introduction to Optimization on Smooth Manifolds
  • Online publication: 09 March 2023
  • Chapter DOI: https://doi.org/10.1017/9781009166164.012
Available formats
×