Linearizations and optimization problems in diffeological spaces

TL;DR

This paper extends linearization and optimization techniques to diffeological spaces, enabling variational methods and convergence analysis in a flexible, infinite-dimensional setting without relying on traditional charts.

Contribution

It introduces a novel framework for optimization in diffeological spaces using generalized linearizations, allowing for smooth path construction and convergence results without canonical charts.

Findings

Constructed smooth paths and variational flows in diffeological spaces.

Developed an optimization algorithm applicable to low-regularity mapping spaces.

Proved weak convergence toward minima under diffeological conditions.

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.