V,
        \label{eq:optimality_conditions:u}\\
        \partial_u {\mathcal{J}}(\bar u,\bar \mu)[v] - a_{\bar\mu}(v,\bar p) &= 0 &&\text{for all } v \in V,
        \label{eq:optimality_conditions:p}\\
        (\nabla_\mu {\mathcal{J}}(\bar u,\bar \mu)+\nabla_{\mu} r^{\textnormal{pr}}_{\bar\mu}(\bar u)[\bar p]) \cdot (\nu-\bar \mu) &\geq 0 &&\text{for all } \nu \in {\mathcal{P}}. 
        \label{eq:optimality_conditions:mu}
        \end{aligned}$$

Note that resembles the state equation . From we deduce the *adjoint – or dual – equation* with unique solution $p_{\mu} \in V$ for a fixed $\mu \in {\mathcal{P}}$, i.e. $$\begin{aligned}
a_\mu(v, p_\mu) = \partial_u {\mathcal{J}}(u_\mu, \mu)[v]
= j_\mu(v) + 2 k_\mu(v, u_\mu)&&\text{for all } v \in V,
\label{eq:dual_solution}\end{aligned}$$ given the solution $u_\mu \in V$ to the state equation . From we observe that the variable $\bar p$ of the optimal triple solves the dual equation for $\bar \mu$. Similarly to the primal solution, we can consider the dual solution map $\mathcal A:{\mathcal{P}}\to V$, $\mu \mapsto \mathcal A(\mu) := p_\mu$, where $p_\mu$ is the solution of for the parameter $\mu$. In particular, $\bar p = p_{\bar \mu}$. For given $u, p \in V$, we also introduce the dual residual $r_\mu^{\textnormal{du}}(u, p) \in V'$ associated with by $$\begin{aligned}
r_\mu^{\textnormal{du}}(u, p)[v] := j_\mu(v) + 2k_\mu(v, u) - a_\mu(v, p)&&\text{for all }v \in V.
\label{eq:dual_residual}\end{aligned}$$ In addition, from the dual equation , we obtain the 