An Abstract Maximum Principle for constrained minimum problems

TL;DR

This paper introduces the Abstract Maximum Principle, a foundational concept for deriving necessary conditions in constrained optimization and control problems, emphasizing a geometric and topological perspective.

Contribution

It presents a pedagogical formulation of the Abstract Maximum Principle, connecting it to existing set-separation approaches for necessary optimality conditions.

Findings

Derives necessary conditions for finite-dimensional minima.

Applies to optimal control problems.

Highlights geometric and topological insights in optimization.

Abstract

This article makes no claim to originality, other than, perhaps, the simple statement here called the {\it Abstract Maximum Principle}. Actually, the whole contents are strongly based on some H. Sussmann's and coauthors' papers, in which, in a much more general context, the set-separation approach is regarded as foundational for necessary conditions for minima. So, rather than being the exposition of original material, this paper has mainly a pedagogical purpose. From the Abstract Maximum Principle it is possible to deduce several necessary conditions for both finite dimensional minimum problems and for optimal control problems. More in general, this Principle seems apt to capture some consequences of the geometric and topological idea of (possibly vector-valued) minimization in a parametrized problem.