Legendre–Fenchel transformation

In mathematics and mathematical optimization, the convex conjugate of a function is a generalization of the Legendre transformation which applies to non-convex functions. It is also known as Legendre–Fenchel transformation or Fenchel transformation (after Adrien-Marie Legendre and Werner Fenchel). It is used to transform an optimization problem into its corresponding dual problem, which can often be simpler to solve.

Let ${\displaystyle X}$ be a real topological vector space, and let ${\displaystyle X^{*}}$ be the dual space to ${\displaystyle X}$. Denote the dual pairing by

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