Let $X,Y$ be normed vector spaces. Let $f$ be a linear continuous functional and $G:X\to Y$ is linear. By Kuhn-Tucker theorem, if $x_0$ minimize $f(x)$ subject to $G(x)\le 0$, then we can find $y^*\in Y^*,y^*\ge 0$ such that the Lagrangian
$$L(x,y^{*}) = f(x) + \langle G(x),y^{*} \rangle$$
is stationary at $x_0$ and $\langle G(x_0), y^{*} \rangle = 0$. For any $y^*\ge y^*_0\ge 0$, Do we have $x\mapsto \langle G(x),y^*_0\rangle$ also continuous?