conjugate

The Legendre transform is an self-inverse transformation on real-valued convex functions of one real variable. The generalization of the Legendre transformation to affine spaces and non-convex functions is known as the convex conjugate (also called the Legendre–Fenchel transformation), which can be used to construct a function’s convex hull. Definition Let $f$ be a convex function. The Legendre transform of $f$ is defined as $$f^*(y)=\sup_x y^T x-f(x),$$ which is always well-defined when $f(x)$ is convex....