following formulation for the dual sensitivities.

\[prop:dual\_solution\_dmu\_eta\] Considering the dual solution map $\mathcal A:{\mathcal{P}}\to V$, $\mu \mapsto p_\mu$, its directional derivative $d_{\eta} p_\mu\in V$ w.r.t. a direction $\eta \in {\mathcal{P}}$ is the solution of $$\label{eq:dual_sens}
    \begin{split}
    a_\mu(q, d_{\eta} p_\mu) &= -\partial_\mu a_\mu(q, p_\mu)\cdot \eta + d_\mu \partial_u {\mathcal{J}}(u_\mu, \mu)[q] \cdot \eta        = \partial_\mu r_\mu^{\textnormal{du}}(u_\mu, p_\mu)[q] \cdot \eta + 2 k_\mu(q, d_{\eta}u_\mu)
    \end{split}$$ for all $q \in V$, where the latter equality holds for quadratic ${\mathcal{J}}$ as in .

Note that $\mathcal A$ is well defined because the bilinear form $a_{\mu}(\cdot \, , \cdot)$ is continuous and coercive. For a proof of the other claims we refer to [@HPUU2009], for instance.

Furthermore, we can compute first-order derivatives of ${\hat{\mathcal{J}}}$.

\[prop:grad\_Jhat\] For given $\mu\in{\mathcal{P}}$, the gradient of ${\hat{\mathcal{J}}}$, $\nabla_\mu{\hat{\mathcal{J}}}: {\mathcal{P}}\to {\mathbb{R}}^P$, is given by $$\begin{aligned}
    \nabla_{\mu}{\hat{\mathcal{J}}}(\mu) &= \nabla_{\mu}{\mathcal{J}}(u_{\mu}, \mu)+\nabla_{\mu}r_\mu^{\textnormal{pr}}(u_{\mu})[p_{\mu}]= \nabla_\mu \Theta(\mu) + \nabla_\mu j_\mu(u_\mu)
    +  \nabla_\mu k_\mu(u_\mu, u_\mu) + \nabla_\mu l_\mu(p_\mu) - \nabla_\mu a_\mu(u_\mu, p_\mu).
    \end{aligned}$$

This follows from , , and , cf. [@HPUU2009].

The proof of Proposition [[\[prop:grad\_Jhat\]]{}]{} relies on the fact that both $u_\mu$ and $p_\mu$ belong to the same space $V$; cf. *[@HPUU2009]*. In particular, for any $\mu\in{\mathcal{P}}$, we have $\nabla_\mu {\hat{\mathcal{J}}}(\mu) = \nabla_\mu \mathcal L(u_\mu,\mu,p_\mu)$.

For $\bar\mu$ 