Given a convex body $K$ and a point $y$ outside the convex body (in the ambient space), the Bregman projection of $y$ , with respect to the regularizer $R$, is defined as
$x=\rm{argmin}\{B_{R}\left(\omega, y\right):\omega \in K\}$
where, $B_{R}\left(\omega, y\right)=R\left(\omega\right)+R\left(y\right)-\nabla R\left(y\right)'\left(\omega-y\right)$
Then how can I show the following
$[\nabla R(x) − \nabla R(y)]' (\omega − x) >0$
Please give me some hint.
Recall that the subdifferential of the indicator function is the normal cone. Take the optimality conditions,
$$0\in N_K(x) + \nabla R(x) - \nabla R(y)\implies \nabla R(y) - \nabla R(x) \in N_K(x)$$
where $N_K$ is the normal cone. By definition of the normal cone you are done.