solution as well as their sensitivities (see [@HAA2017]) or for deriving optimality conditions for (see [@HPUU2009]).

\[prop:solution\_dmu\_eta\] Considering the solution map $\mathcal S:{\mathcal{P}}\to V$, $\mu \mapsto u_\mu$, its Fréchet derivative $d_{\nu} u_\mu \in V$ w.r.t. a direction $\nu\in\mathbb{R}^P$ is the unique solution of $$\begin{aligned}
\label{eq:primal_sens}
    a_\mu(d_{\nu}u_\mu, v) = \partial_\mu r_\mu^{\textnormal{pr}}(u_\mu)[v] \cdot \nu &&\text{for all } v \in V.
    \end{aligned}$$

We refer to [@HPUU2009] for the proof of this result.

Optimal solution and optimality conditions {#sec:first_order_optimality_conditions}
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In this section, we discuss the existence of an optimal solution for problem . Then, we characterize a locally optimal solution through first- and second-order optimality conditions. Throughout the paper, a bar indicates optimality.

\[Thm:existence\_optimalsolution\] Problem admits an optimal solution pair $(\bar u, \bar \mu)\in V\times{\mathcal{P}}$, where $\bar u:= u_{\bar \mu}$ is the solution of for the parameter $\bar\mu$.

Note that the quantities involved in problem satisfies Assumption 1.44 in [@HPUU2009]. Thus the existence follows from [@HPUU2009 Theorem 1.45].

Let us introduce the reduced cost functional ${\hat{\mathcal{J}}}: {\mathcal{P}}\mapsto \mathbb{R},\,\mu\mapsto {\hat{\mathcal{J}}}(\mu) := {\mathcal{J}}(u_\mu, \mu)= {\mathcal{J}}( \mathcal S(\mu),\mu)$. Then problem is equivalent to the so-called reduced problem $$\begin{aligned}
\min_{\mu \in {\mathcal{P}}} {\hat{\mathcal{J}}}(\mu).
\tag{$\hat{\textnormal{P}}$}\label{Phat}\end{aligned}$$

1.  Since $r^{\textnormal{pr}}_\mu(u_\mu)[p] = 0$ for any $p\in V$, it follows that ${\hat{\mathcal{J}}}(\mu) 