New perspective on some classical results in analysis and optimization

Abstract

The paper illustrates connections between classical results of Analysis and Optimization. The focus is on new elementary proofs of Implicit Function Theorem, Lusternik's Theorem, and optimality conditions for equality constrained optimization problems. The proofs are based on Fermat's Theorem and the Weierstrass Theorem and do not use the contraction mapping principle or other advanced results of Real Analysis, so they can be used in any introductory course on Optimization or Real Analysis without the requirement of the advanced background in analysis. The paper also presents a simple proof of Implicit Function Theorem in normed linear spaces.