**Convex conjugate functions and Fenchel’s inequality**

Let be some function. The **convex conjugate** of (also know as the **Fenchel transformation**), is the function defined as

**Fenchel’s equality** is the following statement:

**Theorem (Fenchel’s inequality).** For any ,

The proof of Fenchel’s equality follows directly from the definition of the convex conjugate:

A direct application of Fenchel’s inequality shows that the conjugate of the conjugate (the **biconjugate**), denoted , always satisfies

To see this: Fenchel’s inequality says for all . Taking a supremum over all , the inequality becomes

It turns out that if is closed and convex, then we actually have . The proof is a bit cumbersome and so I’m not providing it in this post. (For a proof, see Reference 1 or Slides 5.13-5.14 of Reference 2.) More generally, the **Fenchel-Moreau theorem** gives necessary and sufficient conditions for .

**Subdifferentials and optimality**

A vector is a **subgradient** of at point if

for all .

The **subdifferential** of at point , denoted , is the set of all the subgradients of and point .

Convex conjugates appear in the following theorem concerning subdifferentials:

**Theorem.** If is closed and convex, then for any ,

*Proof (adapted from Slide 5.15 of Reference 2 and Reference 3):* First we show that . Using the definitions of subgradients and convex conjugates,

Thus, for any ,

which implies that .

To get the reverse implication , apply the same logic above with in place of and use the fact that for closed convex .

Next, we show that . Using the definitions of subgradients,

Taking supremum over all , the inequality becomes . Combining this with Fenchel’s inequality gives us equality .

Conversely, if , we have

References:

- StackExchange. “How to prove the conjugate of the conjugate function is itself?”
- Vandenberghe, L. (2022). “5. Conjugate functions.”
- StackExchange. “Proof about Conjugate and subgradient.”