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Fenchel conjugate

The convex conjugate of an extended-real-valued function, defined by a supremum of affine functionals.

A Fenchel conjugate of an extended-real-valued f:Rn→(−∞,+∞]f:\mathbb{R}^n\to(-\infty,+\infty] is the function f∗:Rn→(−∞,+∞]f^*:\mathbb{R}^n\to(-\infty,+\infty] defined by

f∗(y)  =  sup⁡x∈Rn(⟨y,x⟩−f(x)),y∈Rn,f^*(y) \;=\; \sup_{x\in\mathbb{R}^n}\big(\langle y,x\rangle - f(x)\big), \qquad y\in\mathbb{R}^n,

where ⟨y,x⟩\langle y,x\rangle denotes the Euclidean pairing.

This operation (also called the ) turns ff into a (possibly extended-real) , and it is the basic object behind and . The definition uses the over all affine functionals x↦⟨y,x⟩−f(x)x\mapsto \langle y,x\rangle - f(x).

Examples
  • If f(x)=12∥x∥22f(x)=\tfrac12\|x\|_2^2 on Rn\mathbb{R}^n, then f∗(y)=12∥y∥22f^*(y)=\tfrac12\|y\|_2^2.
  • If f=δCf=\delta_C is the indicator of a nonempty set CC (i.e. δC(x)=0\delta_C(x)=0 for x∈Cx\in C and +∞+\infty otherwise), then f∗(y)=sup⁡x∈C⟨y,x⟩f^*(y)=\sup_{x\in C}\langle y,x\rangle, the support function of CC.