A simple proof of the Karush-Kuhn-Tucker theorem with finite number of equality and inequality constraints

TL;DR

This paper presents a straightforward proof of the Karush-Kuhn-Tucker theorem for problems with finite constraints, utilizing basic linear algebra and the local inverse theorem to simplify understanding.

Contribution

It offers a concise and elementary proof of the KKT theorem, making the foundational result more accessible and easier to understand.

Findings

Proof relies on elementary linear algebra lemma

Uses local inverse theorem for simplicity

Provides a shorter, clearer proof of KKT conditions

Abstract

We provide a simple and short proof of the Karush-Kuhn-Tucker theorem with finite number of equality and inequality constraints. The proof relies on an elementary linear algebra lemma and the local inverse theorem.