Consider a convex optimization problem.
$$\min_{u\in\Re^k} f(u)$$
s.t. $g_i(u)\leq0,\ i=1,\ldots,m$
Let $F(x)=F(u,\lambda)=(f'(u)+\sum_{i=1}^m\lambda_ig_i'(u),-g_1(u),\ldots,-g_m(u)):\Re^n\rightarrow\Re^n$, ($n=k+m$)
$G=\{x=(u,\lambda)\in\Re^n:\lambda\geq0\}$.
$f$ and $g_i$ are all convex. For simplicity, assume they belong to $C^1$.
Is $F(x)$ a monotone operator in $G$, i.e., $(F(x_1)-F(x_2))^T(x_1-x_2)\geq0$ always satisfied in G?
Thanks a lot.