DefinitionDefinition: Literature-sourcedProof: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

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Fenchel conjugate of an extended-real function

Definition

Let $f : R^{n} \to (- \infty, + \infty]$ . Its Fenchel conjugate is the extended-real function on the same Euclidean space $f^{*} (s) : = \sup_{x \in R^{n}} {⟨ s, x ⟩ - f (x)} .$ When $f (x) = + \infty$ , the corresponding term is $- \infty$ and does not increase the supremum. Its effective domain is $dom f^{*} = {s : f^{*} (s) < + \infty}$ .

Facts & Assumptions

Given: $n \in N$ , $f : R^{n} \to (- \infty, + \infty]$ , and the Euclidean pairing.

[A1]

Function values lie in $(- \infty, + \infty]$ and the pairing is Euclidean.

[F1]

Proper closed convex extended-real functions on $R^{n}$ : the domain of a proper function is nonempty, and its values lie in $(- \infty, + \infty]$ .

[F2]

The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ : $⟨ s, x ⟩$ is a finite real number for $s, x \in R^{n}$ .

Proof

1.1A1givenF1F2given

For every $x$ with finite $f (x)$ , the expression $⟨ s, x ⟩ - f (x)$ is real; for $f (x) = + \infty$ it contributes $- \infty$ under the stated convention. Thus the supremum is unambiguously defined in $[- \infty, + \infty]$ .

2.1A1F1step 1.1choose∎

If $f$ is proper, choose one $x_{0} \in dom f$ . The finite term $⟨ s, x_{0} ⟩ - f (x_{0})$ shows $f^{*} (s) > - \infty$ for every $s$ ; it may still be $+ \infty$ . This definition alone does not assert finiteness everywhere.

Depends on

Used by

Dependency tree · two levels

18 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Stephen Boyd and Lieven Vandenberghe, Convex Optimization, §3.3.1 (standard reference, not scraped)
Neal Parikh and Stephen Boyd, Proximal Algorithms, §1.2 (standard reference, not scraped)