Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Linearizations and optimization problems in diffeological spaces

Jean-Pierre Magnot

Abstract

By generalizing the notion of linearization, a concept originally arising from microlocal analysis and symbolic calculus, to diffeological spaces, we make a first proposal setting for optimization problems in this category. We show how linearizations allow the construction of smooth paths and variational flows without requiring canonical charts or gradients. With these constructions, we introduce a general optimization algorithm adapted to diffeological spaces under weakened assumptions. The method applies to spaces of mappings with low regularity. Our results show that weak convergence toward minima or critical values can still be achieved under diffeological conditions. The approach extends classical variational methods into a flexible, non-linear infinite-dimensional framework. Preliminary steps to the search for fixed points of diffeological mappings are discussed.

Linearizations and optimization problems in diffeological spaces

Abstract

By generalizing the notion of linearization, a concept originally arising from microlocal analysis and symbolic calculus, to diffeological spaces, we make a first proposal setting for optimization problems in this category. We show how linearizations allow the construction of smooth paths and variational flows without requiring canonical charts or gradients. With these constructions, we introduce a general optimization algorithm adapted to diffeological spaces under weakened assumptions. The method applies to spaces of mappings with low regularity. Our results show that weak convergence toward minima or critical values can still be achieved under diffeological conditions. The approach extends classical variational methods into a flexible, non-linear infinite-dimensional framework. Preliminary steps to the search for fixed points of diffeological mappings are discussed.

Paper Structure

This paper contains 21 sections, 13 theorems, 72 equations, 1 algorithm.

Table of Contents

  1. Preliminaries on diffeologies
  2. Diffeologies and Frölicher spaces
  3. Groups, vector spaces, algebra and fiber bundles in diffeologies
  4. Linearizations from Finite-Dimensional Manifolds to Diffeological Spaces and Infinite-Dimensional Settings
  5. Linearizations of a Manifold
  6. Exponential Map and Linearization on a Compact Riemannian Manifold
  7. Properties of the exponential map
  8. The Exponential Map Defines a Linearization in the Sense of Bokobza-Haggiag
  9. Extending Classical Optimization Methods in Linearizations
  10. A Generalized Notion of Linearization
  11. Distance Induced by a Linearization
  12. Gradient and steepest descent methods... (Almost) Without gradient
  13. The Algorithm Associated to a Method $\mathcal{M}$
  14. Classical A Priori Convergence Results to Weak Solutions
  15. Application to Mapping Spaces with Low Regularity
  16. ...and 6 more sections

Key Result

Proposition 5

Let $(X, \mathcal{F}, \mathcal{C})$ and $(X', \mathcal{F}', \mathcal{C}')$ be Frölicher spaces. A map $f: X \to X'$ is smooth in the Frölicher sense if and only if it is smooth for the associated diffeologies $\mathcal{P}_\infty(\mathcal{F})$ and $\mathcal{P}_\infty(\mathcal{F}')$. $\blacktrianglele

Theorems & Definitions (39)

  • Definition 1
  • Definition 2
  • Example 3
  • Definition 4
  • Proposition 5: Ma2006-3
  • Remark 6
  • Definition 7: Wa
  • Example 8
  • Proposition 9: IgdiffSou
  • Proposition 10
  • ...and 29 more