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Fenchel–Young inequality and its equality case

Statement

Let $f : R^{n} \to (- \infty, + \infty]$ be proper and convex. For every $x, s \in R^{n}$ , $f (x) + f^{*} (s) \geq ⟨ s, x ⟩,$ with extended-real interpretation. Equality holds exactly when $x \in dom f$ , $f^{*} (s) < + \infty$ , and $s \in \partial f (x)$ ; equivalently, for proper convex $f$ , equality itself forces the two finiteness conditions and the subgradient membership.

Facts & Assumptions

Given: Proper convex $f$ , $x, s \in R^{n}$ .

[A1]

The function $f$ is proper and convex.

[F1]

Fenchel conjugate of an extended-real function: $f^{*} (s) = \sup_{z} (⟨ s, z ⟩ - f (z))$ .

[F2]

Subdifferentials of proper extended-real convex functions: for $x \in dom f$ , $s \in \partial f (x)$ iff $f (z) \geq f (x) + ⟨ s, z - x ⟩$ for every $z \in R^{n}$ .

Proof

1.1A1F1given

If $f (x) = + \infty$ , the left side is $+ \infty$ because [F1] and properness ensure $f^{*} (s) > - \infty$ , so the inequality is true and equality is impossible. If $f (x) < + \infty$ , the term $z = x$ in [F1] gives $f^{*} (s) \geq ⟨ s, x ⟩ - f (x)$ , proving the inequality; if $f^{*} (s) = + \infty$ , equality is again impossible.

2.1A1F1F2step 1.1algebra∎

Assume now $f (x)$ and $f^{*} (s)$ are finite. Equality in step 1.1 holds iff the supremum in [F1] equals its $z = x$ term, which is equivalent to $⟨ s, z ⟩ - f (z) \leq ⟨ s, x ⟩ - f (x) for every z \in dom f .$ Rearranging gives precisely the inequalities in [F2]; for $z \notin dom f$ they hold automatically. Thus equality is equivalent to $s \in \partial f (x)$ , and the converse follows by reversing these inequalities.

Depends on

Used by

Two-block ADMM residual, objective and dual sequence convergence Theorem

Dependency tree · two levels

6 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)