Convex Analysis and Duality

TL;DR

This paper explores the fundamental role of convexity and duality in nonlinear optimization and functional analysis, deriving powerful tools from elementary duality principles with applications to variational problems.

Contribution

It introduces simple, powerful duality-based tools derived from elementary principles for analyzing convexity in optimization and functional analysis.

Findings

Derived new tools from duality arguments for convex analysis

Applied duality methods to variational problems

Highlighted the importance of elementary duality in optimization

Abstract

Convexity is an important notion in non linear optimization theory as well as in infinite dimensional functional analysis. As will be seen below, very simple and powerful tools will be derived from elementary duality arguments (which are byproducts of the Moreau-Fenchel transform and Hahn Banach Theorem). We will emphasize on applications to a large range of variational problems. Some arguments of measure theory will be skipped.