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Optimization on Weak Riemannian Manifolds

Valentina Zalbertus, Max Pfeffer, Alexander Schmeding

Abstract

Riemannian structures on infinite-dimensional manifolds arise naturally in shape analysis and shape optimization. These applications lead to optimization problems on manifolds which are not modeled on Banach spaces. The present article develops the basic framework for optimization via gradient descent on weak Riemannian manifolds leading to the notion of a Hesse manifold. Further, foundational properties for optimization are established for several classes of weak Riemannian manifolds connected to shape analysis and shape optimization.

Optimization on Weak Riemannian Manifolds

Abstract

Riemannian structures on infinite-dimensional manifolds arise naturally in shape analysis and shape optimization. These applications lead to optimization problems on manifolds which are not modeled on Banach spaces. The present article develops the basic framework for optimization via gradient descent on weak Riemannian manifolds leading to the notion of a Hesse manifold. Further, foundational properties for optimization are established for several classes of weak Riemannian manifolds connected to shape analysis and shape optimization.

Paper Structure

This paper contains 13 sections, 25 theorems, 92 equations, 2 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Weak Riemannian Manifolds in Optimization
  4. Optimality Conditions
  5. First-Order Optimality Conditions
  6. Second-Order Optimality Conditions
  7. The Riemannian Gradient Descent Method
  8. Classes of Hesse manifolds and their Optimization-relevant properties
  9. Robust Riemannian manifolds
  10. Strong Riemannian manifolds
  11. Computation of the Riemannian gradient and the Riemannian Hessian
  12. Numerical Experiments
  13. Sprays, connections and metrics

Key Result

Theorem 1.1

Let $f\colon M\to \mathbb{R}$ be continously differentiable on a weak Riemannian manifold $M$. Then every local minimizer $p\in M$ satisfies where $\nabla f$ denotes Riemannian gradient of $f$.

Figures (2)

  • Figure 1: Riemannian gradient descent for $f$. Left: evolution of the iterates. Right: function values and gradient norms over twenty iterations.
  • Figure 2: Riemannian gradient descent for $f_g$. Left: evolution of the iterates. Right: functional values and gradient norms over twenty iterations.

Theorems & Definitions (77)

  • Theorem 1.1: First-Order Optimality
  • Theorem 1.2
  • Theorem 1.3: Second-Order Optimality
  • Theorem 1.4
  • Definition 2.1: Weak/Strong Riemannian Manifold
  • Remark 2.2
  • Definition 2.3: Riemannian Gradient
  • Definition 2.4: Riemannian Hessian
  • Example 3.1
  • Remark 3.2
  • ...and 67 more