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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⟩=∑i=1nyixi\langle y,x\rangle=\sum_{i=1}^n y_i x_i is the Euclidean .

Properties and limiting cases

If f≡+∞f\equiv+\infty, then f∗≡−∞f^*\equiv-\infty. Otherwise ff is finite somewhere and f∗f^* never takes the value −∞-\infty.

When ff is finite somewhere, its conjugate is a (possibly extended-real-valued) , because it is a pointwise of affine functions of yy. It is also called the .

Examples
  • If f(x)=12∥x∥22f(x)=\tfrac12\lVert x\rVert_2^2, then f∗(y)=12∥y∥22f^*(y)=\tfrac12\lVert y\rVert_2^2.
  • If f=δCf=\delta_C is the indicator of a nonempty set CC, then f∗(y)=sup⁡x∈C⟨y,x⟩f^*(y)=\sup_{x\in C}\langle y,x\rangle, the support function of CC.