The Legendre Transformation in Modern Optimization

The Legendre transformation (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. Application of LET to a strictly convex and smooth function leads to Legendre identity (LEID). For strictly convex and tree times differentiable function LET leads to the Legendre invariant (LEINV). Although LET has been known for more then 200 years both LEID and LEINV are critical in modern optimization theory and methods. The purpose of the paper (survey) is to show the role LEID and LEINV play in both constrained and unconstrained optimization.