Optimization on diffeological spaces

Abstract

On this poster, we present optimization techniques on diffeological spaces. Diffeological spaces firstly introduced by J.M. Souriau in the 1980s are a natural generalization of smooth manifolds. In order to generalize optimization methods known on manifolds to diffeological spaces, we define various objects like a diffeological tangent space, a diffeological Riemannian space as well as a diffeological gradient. In addition we give the definition of a diffeological retraction. These objects are necessary for formulating the steepest descent method on diffeological spaces. We present this method and apply it on an example.

On the geometry of diffeological vector pseudo-bundles and infinite dimensional vector bundles: automorphisms, connections and covariant derivatives

“…This defines the internal tangent cone first described for Frölicher spaces in [12]. This approach seems sufficient for application purpose [19].…”