I refer to the following paper: (fix the wrong link) https://papers.nips.cc/paper/5723-adaptive-primal-dual-splitting-methods-for-statistical-learning-and-image-processing.pdf
In that paper, we consider the saddle point problem: $$ \min_x \max_y f(x)+y^TAx - g(y) $$
Then, this paper says that variational inequality formulation is that $$ h(u) - h(u^*) + (u-u^*)^TQ(u^*)\geq 0, \quad\forall u\in\Omega \tag{18} $$
where $h(u) = f(x)+g(y)$ and $u=(x,y)$.
However, I am very confused after reading following material in 62page.
http://meeting.xidian.edu.cn/workshop/miis2012/uploads/files/20120409/20120409170826.pdf
I don't know why $h(u)$, instead of $\nabla h(u)$, appears in (18). Is there anyone who help me?
(added) What I understood is that the solution $x^*$ of the following problem for a convex set $K$, $$ \min_x f(x) \quad \text{s.t. } x\in K $$ satisfy the following: $$ (y-x^*)^T\nabla f(x^*)\geq 0,\quad \forall y\in K $$
So I wonder why $(u-u^*)\nabla h(u^*)$ doesn't appear in (18).