Here is a lemma that I find myself googling regularly since I always forget it’s exact form.

Lemma 1Let be a monotone operator, and denote by the resolvent of . Then it holds that

*Proof:* We start with the left hand side and deduce

I do not know any official name of this, but would call it *Moreau’s identity* which is the name of the respective statement for proximal operators for convex functions and :

The version for monotone operators is Proposition 23.18 in the first edition of Bauschke and Combette’s book “Convex Analysis and Monotone Operator Theory in Hilbert Spaces”.

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February 24, 2017 at 11:29 pm

Moreau’s identity can also be written as x = R_A(x) + R_Ainv (x) (assuming lambda is 1 for simplicity). This way of writing it emphasizes that Moreau’s identity is a generalization of the linear algebra fact that x = P_W (x) + P_Wperp (x), where W is a subspace of a finite dimensional inner product space and Wperp is the orthogonal complement of W.

February 25, 2017 at 12:39 am

That’s correct – I mainly wrote this down because I always forget the scaling in the . Unfortunately this does not suggest itself from the projection formulation…

March 1, 2017 at 6:50 pm

Does this also hold for general operators A? I didn’t see where you used the monotonicity …

March 1, 2017 at 8:56 pm

That’s true, you only need that the resolvents exist.

March 2, 2017 at 8:37 pm

Dirk, you must remove the parentheses around (I – λ prox) … or next time you code a primal-dual algorithm, you’ll get headache.

March 2, 2017 at 10:36 pm

Oh yeah, thanks!