**Relations and resolvents**

A * relation* on is a subset of . We use the notation to mean the set . You can think of as an operator that maps vectors to sets . (Along this line of thinking, functions are a special kind of relation where every vector is mapped to a set consisting of exactly one element.)

The * inverse relation* is defined as .

For a relation and some parameter , the * resolvent* of is defined as the relation

In other words,

**Connection with the proximal operator**

Let be some convex function. Recall that the * proximal operator* of is defined by

It turns out that * the resolvent of the subdifferential operator is the proximal operator*. Here is the proof: Let for some . By definition of the resolvent,

(* Note:* Setting , the chain of reasoning above gives , which is useful to know.)

References:

- Pilanci, M. (2022). “Monotone Operators.”