I'm reading this pdf while learning about subdifferential calculus, and I came across this expression (page 176):
$$0 \in \partial f(\hat y) + (\hat y - x)/\tau \tag{1}$$
I don't know how to interpret it in terms of the definition of the subdifferential of a convex function $f$. I think it means something like:
$$0 \in \partial f(\hat y) + (\hat y - x)/\tau = \{ p \in \text{Dom} f; f(y) \geq f(\hat y) + \langle p, y-\hat y \rangle + (\hat y - x)/\tau ; \forall y \in \text{Dom} f\}$$
But I also can interpret it like this:
$$0 \in \partial \{f(\hat y) + (\hat y - x)/\tau\} = \{ p \in \text{Dom} f; f(y) + (y - x)/\tau\ \geq f(\hat y) + (\hat y - x)/\tau\ + \langle p, y - \hat y \rangle; \forall y \in \text{Dom} f\}$$
where $\hat y = \text{prox}_{\tau f}(x)$. I assume $x$ is fixed.
Or, maybe, this expression should not be interpreted in terms of the definition of the subdifferential, but in terms of its resolvent. The author writes it in the next line: $$\hat y = (I+\tau \partial f)^{-1} (x) \tag{2}$$ This is equivalent to the proximal map of $f$: $$ \hat y = \underset{x \in \text{Dom}f}{argmin} \{f(y) + \frac{1}{2\tau}||y-x||^2\} \tag{3}$$
Here I don't know how to go from $(2)$ to $(3)$, or how to go from $(1)$ to $(2)$.
I have the Rockafellar's Convex Analysis book from 1972 and I couldn't find how to interpret the expression in $(1)$. I don't have the 1997 edition, so I don't know if more content was added.
The expression in (1) uses the so-called Minkowski sum of two sets $$A + B := \{a + b \mid a \in A, b \in B\}$$ and one of these sets is actually a singleton: $$A + z := \{a + z \mid a \in A\}.$$
The relation between (1) and (2) can be seen by rearranging (1) as $$x = \hat y + \tau \, \partial f(\hat y)$$ and this is equivalent to (2) by definition of the resolvent.
Further, (1) is the optimality condition of the optimization problem in (3) which can be seen by differentiation of the objective of (3) w.r.t. $y$.