Given a PDE-constrained optimization as \begin{equation} \min_{u\in U} J(y,u)=\frac{1}{2}\|y-y_d\|^2_{L^2(\Omega)}+\frac{\alpha}{2}\|u-u_d\|^2_{L^2(\Omega)} \end{equation} subject to \begin{equation} \begin{cases} &-\Delta y=f+u, \text{in }\Omega,\\ &y=0, \text{on }\partial\Omega, \end{cases} \end{equation} Let $y_\tau$ be the discretized solution, $p$ be the adjoint solution associated with $y$ and $p_\tau$ is the discretized solution as well. Define $e_y=y_\tau-y$ and $e_p=p-p_\tau$. The following 'dual' problem is introduced. \begin{equation} \min_{q\in U} J(\phi,q)=\frac{1}{2}\|\phi-e_y\|^2_{L^2(\Omega)}+\frac{\alpha}{2}\|q\|^2_{L^2(\Omega)} \end{equation} subject to \begin{equation} \begin{cases} &-\alpha^{1/2}\Delta \phi=e_p+q, \text{in }\Omega,\\ &\phi=0, \text{on }\partial\Omega, \end{cases} \end{equation}
It is not clear to me how this dual problem is formulated. Especially it mixed up the continuous and discrete space together. This is introduced in the paper section 3.2.