The Legendre Transform in Modern Optimization

TL;DR

This paper surveys the role of the Legendre transform, Legendre identity, and Legendre invariant in modern optimization, highlighting their importance in both constrained and unconstrained problems.

Contribution

It provides a comprehensive overview of how the Legendre transform and its related identities are applied in contemporary optimization theory and methods.

Findings

Legendre transform underpins duality principles in optimization.

Legendre identity and invariant are critical in constrained and unconstrained optimization.

The survey emphasizes the historical and modern significance of these concepts.

Abstract

The Legendre transform (LET) is a product of a general duality principle: any smooth curve is, on the one hand, a locus of pairs, which satisfy the given equation and, on the other hand, an envelope of a family of its tangent lines. An application of the LET to a strictly convex and smooth function leads to the Legendre identity (LEID). For strictly convex and three times differentiable function the LET leads to the Legendre invariant (LEINV). Although the LET has been known for more then 200 years both the LEID and the LEINV are critical in modern optimization theory and methods. The purpose of the paper (survey) is to show the role of the LEID and the LEINV play in both constrained and unconstrained optimization.