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A subdifferential characterization via Busemann functions and applications to DC optimization on Hadamard manifolds

O. P. Ferreira, D. S. Gonçalves, M. S. Louzeiro, S. Z. Németh, J. Zhu

Abstract

This paper investigates the properties of Busemann functions on Hadamard manifolds and their use in optimization algorithms in Riemannian settings. We present a new Busemann-based characterization of the subdifferential, which is particularly well suited to Riemannian optimization. In the classical Hadamard manifold framework, a subgradient provides a global lower model of a convex function expressed through the inverse exponential map. However, this model may fail to exhibit a useful convexity or concavity structure. By contrast, our characterization yields a concave bounding function by exploiting key properties of Busemann functions. We use this concavity to design and analyze difference-of-convex (DC) optimization methods on Hadamard manifolds. In particular, we reformulate the classical DC algorithm (DCA) for Riemannian contexts and study its convergence properties. We also report preliminary numerical experiments comparing the proposed Busemann DCA, which leads to geodesically convex subproblems, with the classical Riemannian DCA.

A subdifferential characterization via Busemann functions and applications to DC optimization on Hadamard manifolds

Abstract

This paper investigates the properties of Busemann functions on Hadamard manifolds and their use in optimization algorithms in Riemannian settings. We present a new Busemann-based characterization of the subdifferential, which is particularly well suited to Riemannian optimization. In the classical Hadamard manifold framework, a subgradient provides a global lower model of a convex function expressed through the inverse exponential map. However, this model may fail to exhibit a useful convexity or concavity structure. By contrast, our characterization yields a concave bounding function by exploiting key properties of Busemann functions. We use this concavity to design and analyze difference-of-convex (DC) optimization methods on Hadamard manifolds. In particular, we reformulate the classical DC algorithm (DCA) for Riemannian contexts and study its convergence properties. We also report preliminary numerical experiments comparing the proposed Busemann DCA, which leads to geodesically convex subproblems, with the classical Riemannian DCA.
Paper Structure (25 sections, 22 theorems, 121 equations, 5 tables, 2 algorithms)

This paper contains 25 sections, 22 theorems, 121 equations, 5 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Notation
  3. Basics results about Hadamard manifolds
  4. Basic results on the $\kappa$-hyperbolic space form
  5. Basic results on the manifold of symmetric positive definite matrices
  6. Subdifferential of a convex function on Hadamard manifolds
  7. The Busemann functions on Hadamard manifolds
  8. Examples of Busemann functions
  9. Characterization for the subdifferential via Busemann functions
  10. The Busemann DC Algorithm for DC optimization
  11. The Busemann DC algorithm
  12. Convergence analysis
  13. Numerical experiments
  14. Test problems in hyperbolic space
  15. DC problem with a $\theta$-order hyperbolic Rosenbrock-type objective
  16. ...and 10 more sections

Key Result

Lemma 1

Let ${\mathbb M}$ be a Hadamard manifold. The following inequality holds: As a consequence, Moreover, if the sectional curvature of ${\mathbb M}$ is identically zero, then both inequalities eq:coslaw2 and eq:log-norm-ineq hold as equalities.

Theorems & Definitions (50)

  • Lemma 1
  • Definition 2
  • Definition 3
  • Proposition 4
  • Lemma 5
  • Definition 6
  • Proposition 7
  • Theorem 8
  • Proposition 9
  • Remark 1
  • ...and 40 more