Knowlpedia

Biconjugate

The Fenchel conjugate of a function's conjugate, central to lower-semicontinuous convex relaxation.

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

f∗∗=(f∗)∗,f^{**}=(f^*)^*,

where f∗f^* is the of ff.

If ff is proper and has an affine minorant, then f∗∗f^{**} is the greatest lower-semicontinuous convex function bounded above by ff. In particular, f∗∗≤ff^{**}\le f pointwise. The states that a proper function satisfies f=f∗∗f=f^{**} exactly when it is lower-semicontinuous and convex.

Here “lower-semicontinuous and convex” is also commonly called .

Examples
  • If ff is proper, lower-semicontinuous, and convex—for instance, a norm—then f∗∗=ff^{**}=f.
  • If C⊆RnC\subseteq\mathbb{R}^n is nonempty and δC\delta_C is its indicator function, then δC∗∗=δconv⁡‾(C)\delta_C^{**}=\delta_{\overline{\operatorname{conv}}(C)}. For C={−1,1}⊂RC=\{-1,1\}\subset\mathbb{R}, this gives δC∗∗=δ[−1,1]\delta_C^{**}=\delta_{[-1,1]}.