*state – or primal – equation* $$\begin{aligned}
    a_\mu(u, v) = l_\mu(v) && \text{for all } v \in V,
    \tag{P.b}\label{P.state}
    \end{aligned}$$

where, for each $\mu \in {\mathcal{P}}$, $a_\mu: V \times V \to {\mathbb{R}}$ denotes a continuous and coercive symmetric bilinear form and $l_\mu \in V'$ denotes a continuous linear functional. For given $u \in V$, $\mu \in {\mathcal{P}}$, we introduce the primal residual $r_\mu^{\textnormal{pr}}(u) \in V'$ associated with by $$\begin{aligned}
r_\mu^{\textnormal{pr}}(u)[v] := l_\mu(v) - a_\mu(u, v) &&\text{for all }v \in V.
\label{eq:primal_residual}\end{aligned}$$ The primal residual plays a crucial role for a posteriori error analysis and for sensitivities of solution maps.

The Lagrange functional for is given by $\mathcal{L}(u,\mu,p) = {\mathcal{J}}(u,\mu) + r_\mu^{\textnormal{pr}}(u)[p]$ for $(u,\mu)\in V\times{\mathcal{P}}$ and for $p\in V$.

To apply RB methods efficiently, we require the parametrization of the problem to be separable from $V$ throughout the work. This separability is a standard assumption for RB methods and can be circumvented by using empirical interpolation techniques [@BMNP2004; @CS2010; @DHO2012].

\[asmpt:parameter\_separable\] We assume $a_\mu$, $l_\mu$, $j_\mu$, $k_\mu$ to be parameter separable with $\Xi^a, \Xi^l, \Xi^j, \Xi^k \in {\mathbb{N}}$ non-parametric components $a_\xi: V \times V \to {\mathbb{R}}$ for $1 \leq \xi \leq \Xi^a$, $l_\xi \in V'$ for $1 \leq \xi \leq \Xi^l$, 