\section{Additional Illustration}\label{sec:additional_figure}
We put the illustration of a 1D Toy Example in \figref{fig:1D_illustration}.
\begin{figure}
    \centering
            {
    \includegraphics[trim={1cm 1.0cm 1cm 4cm},
        width=.8\textwidth]{./fig/EG1d_vertical.pdf}
    }
    \caption{Illustration of \algname on a synthetic noise-free 1D example. The first two rows show the GP for the $\cFunc$, the superlevel-set $S_\cFunc$, the region of interest $\roi_{\cFunc}$ and the corresponding acquisition function $\acqC(\instance)$ as defined in \eqref{eq:acqC}. The following two rows show the GP for $\globalf$, the region of interest $\roi_{\globalf}$, and the corresponding acquisition function $\acqF(\instance)$ defined in \eqref{eq:acqF}. We show that after identifying $S_\cFunc$, we could define the threshold for ROI identification of $\globalf$ accordingly.
    The bottom row demonstrates that the general ROI  $\roi$ as defined in \eqref{eq:roi} is identified by taking the intersection ROI for $\globalf$ and $\cFunc$. The general acquisition function is defined as the maximum of the acquisition for $\globalf$ and $\cFunc$ and is maximized on the $\roi$. The scaling and length scale of the GPs are learned via maximum likelihood estimation. }   
    \label{fig:1D_illustration}
\end{figure}
