Let be a convex function. **Moreau’s Decomposition Theorem** is the following result:

**Theorem (Moreau Decomposition).** For all ,

,

where is the **proximal operator** for and is the **convex conjugate** of .

Here is the proof: Let . Then,

The second equivalence is a result involving convex conjugates and subdifferentials (see this post for statement and proof) while the 4th equivalence is a property of the proximal operator (see the note at the end of this post).

**Extended Moreau Decomposition**

The extended version of Moreau’s Decomposition Theorem involves a scaling factor . It states that for all ,

*Proof:* Applying Moreau decomposition to the function gives

Using the definitions of the proximal operator and the convex conjugate,

References:

- Gu, Q. (2016). “SYS 6003: Optimization. Lecture 25.”

### Like this:

Like Loading...

*Related*