Legendre Transform and Lagrange Dual

by allenlu2007

 

* Both of them are defined in convex functions

* Both of them can be generalized to non-convex function but extended to convex hull

* Legendre transform is more general and (sort of) unique   f(x) -> f*(v) -> f(x).   f*(v) is conjugate function.

* Lagrange dual is only for minimum value equivalent; depending on the constraints, Lagrange dual can be more than one.

* Lagrange dual can utilize conjugate function.

 

Young-Fenchel inequality

Fenchel duality theorem

Advertisements