\mu}$ and $p_{{{r}}, \mu}$ denote the primal and dual reduced solutions of and , respectively. To approximate , we introduce the *RB reduced functional* by $$\begin{aligned}
{\hat{J}}_{{r}}(\mu) := {\mathcal{J}}(u_{{{r}}, \mu}, \mu)={\mathcal{J}}(\mathcal S_r(\mu), \mu),&&
\text{where $u_{{{r}}, \mu} \in V_{{r}}^{\textnormal{pr}}$ is the solution of \eqref{eq:state_red}}
\label{eq:Jhat_red}\end{aligned}$$ instead of ${\hat{\mathcal{J}}}_h$ and the problem of finding a locally optimal solution $\bar \mu_{{r}}$ of $$\begin{aligned}
\min_{\mu \in {\mathcal{P}}} {\hat{J}}_{{r}}(\mu).
\label{eq:Phat_red_uncorrected}\end{aligned}$$ Now, a solution to the optimality system is approximated by the RB triple $(u_{{{r}},\bar\mu_{{r}}},\bar\mu_{{r}},p_{{{r}},\bar\mu_{{r}}})$.

As proposed in [@QGVW2017], for computing an approximation of the gradient of ${\hat{J}}_{{r}}$, the gradient from Propostion \[prop:grad\_Jhat\] can be utilized by replacing $u_{\mu}$ and $p_{\mu}$ with their RB counterparts. However, it can not be guaranteed in general that the computed gradient is the actual gradient of ${\hat{J}}_{{r}}$, if $V^{\textnormal{pr}}_{{r}}$ and $V^{\textnormal{du}}_{{r}}$ are chosen to be different. To see this, we consider first the Lagrangian and note that, for $1\leq i\leq P$ and all $p\in V$, it holds $${\hat{J}}_{{r}}(\mu)  = \mathcal{L}(u_{{{r}},\mu},\mu,p), \qquad
    \label{Gradient_lagrangian}
     \big(\nabla_\mu {\hat{J}}_{{r}}(\mu)\big)_i  =  \partial_u \mathcal{L}(u_{{{r}},\mu},\mu,p)[d_{\mu_i}u_{{{r}},\mu}] 
     + d_{\mu_i} \mathcal{L}(u_{{{r}},\mu},\mu,p).$$ Now, following [@QGVW2017], we define the *inexact gradient* ${\widetilde\nabla}_\mu {\hat{J}}_{{r}}:{\mathcal{P}}\to {\mathbb{R}}^P$ by $$\label{naive:red_grad}
\big({\widetilde\nabla}_\mu {\hat{J}}_{{r}}(\mu)\big)_i := d_{\mu_i}{\mathcal{J}}(u_{{{r}}, \mu}, \mu) + d_{\mu_i} r_\mu^{\textnormal{pr}}(u_{{{r}}, \mu})[p_{{{r}}, \mu}] = d_{\mu_i} \mathcal{L}(u_{{{r}},\mu},\mu,p_{{{r}},\mu})$$ for all $1 