I'm reading this thesis on page 182. We work in a Hilbert space $H$, and $y \in H$ solves the variational inequality: $$y \in K : \langle Ay-f, v-y \rangle \geq 0 \; \forall v \in K$$ for some given closed and convex set $K \subset H$.
Define $\lambda = f-Ay \in H^*$ where $f \in H^*$ is given. We have a quantity
$$\delta\lambda \in (R_K(y) \cap \lambda^\perp)^\circ$$
where $C^\circ$ means the polar cone of $C$ and $\lambda^\perp$ is the nullspace of a functional $\lambda$, and $R_K(y)$ is the radial cone of $K$ at the point $y \in K$. We have also that
$$\delta y \in T_K(y)\cap \lambda^\perp$$
where $T_K(y)$ is the tangent cone.
The author states that due to the polyhedricity of $K$, which means that $$\overline{R_K(y)\cap \lambda^\perp} = T_K(y) \cap \lambda^\perp,$$ and because $\delta\lambda = h-A\delta y$, we can deduce that $\delta_y$ solves the variational inequality $$\delta y \in T_K(y) \cap \lambda^\perp : \langle A\delta y - h, v-\delta y \rangle \geq 0, \; \forall v \in T_K(y) \cap \lambda^\perp.$$
I can't get how $\delta y$ satisfies that inequality using the definitions of the polar cone and usual manipulations. I can only get $$\langle h-A\delta y, z \rangle = \langle \delta\lambda, z \rangle \leq 0 \;\forall z \in T_K(y)\cap \lambda^\perp.$$ How can I get the correct element in the dual pairing? I don't see how I can choose $z=c(v-\delta y)$ for $c > 0$ and $v \in T_K(y)\cap \lambda^\perp$ arbitrary. Can someone help with the steps?
The author seems to be a little bit fast at this point: at this point you only get that $\delta\lambda \in (T_K(y) \cap \lambda^\perp)^o$. The next chain of inequalities gives $\langle \delta\lambda, \delta y\rangle = 0$. Then, the VI for the derivative follows.