I have a hard time seeing the relationship between a monotone operator and monotone function.
A monotone function (increasing in this case) in real analysis is defined as:
Let $f:I \mapsto \Bbb R $ where I is an interval of $\Bbb R$ is monotone if: $\forall x,y \in \Bbb R$ $x < y \implies f(x) < f(y)$. The notion of monotony can be extended to that of partially ordered sets (adds more conditions to it, but the spirit of the property remains the same).
A monotone operator on the other hand is defined as:
Let $f:X \mapsto P(X)$ a valued set operator on a hilbert space X.
$f$ is monotone if $\forall (x,y)\in G(f),\forall (x',y')\in G(f)\langle y-y'|x-x'\rangle \geq 0$
Is the second a generalization of the first?
A simple example using $\Bbb R$ with the standard scalar product shows that the two properties are related: let $f:\Bbb R \mapsto\Bbb R$ a function. Let $x, y \in\Bbb R$.
$$\langle f(x) - f(y) | x - y\rangle = (f(x) - f(y))(x-y) \geq 0 \iff \operatorname{sign}(f(x) - f(y)) = \operatorname{sign}(x - y)$$
More generally, take any real valued increasing function and extend it to $R^n$, this equivalence holds up pretty well.
The second definition seems to come from an entirely different field, and I have the feeling that they are not that equivalent (the order relation is not at all used in convex optimization to my knowledge).