Rank theorem in infinite dimension and lagrange multipliers

TL;DR

This paper extends the rank theorem to infinite-dimensional Banach spaces and applies it to derive a Lagrange multiplier rule for constrained optimization problems, with an application to variational problems.

Contribution

It introduces an infinite-dimensional rank theorem and establishes a KKT theorem for Banach space optimization, expanding classical finite-dimensional results.

Findings

Established a rank theorem in infinite dimensions.

Derived a KKT theorem for Banach space optimization.

Applied results to variational problems with constraints.

Abstract

We use an extension to the infinite dimension of the rank theorem of the differential calculus to establish a Karush-Huhn-Tucker theorem for optimization problems in Banach spaces. We provide an application to variational problems on bounded processus under equality constraints.