Fenchel conjugate

Summary

The Fenchel conjugate, or convex conjugate, of a function $f$ on a vector space is the function $f^*$ on the dual space given by $f^*(y) = \sup_x \big(\ip{y}{x} - f(x)\big)$. It is always convex and lower semicontinuous, and for proper convex lower semicontinuous $f$ one has $f^{**} = f$. It extends the classical Legendre transformation to functions that need not be differentiable. It is the basic tool of duality in convex optimization.