\section{Experiments}\label{sec: exps}
\begin{figure*}
  \centering
      {
    \includegraphics[trim={3.3cm 0.5cm 2cm .5cm}, 
    width=.6\textwidth]{./fig/simple_regret_uai_no_config.pdf}
  }
\caption{The input dimensionality, the number of constraints, and the approximate portion of the feasible region in the whole search space for each task are denoted on the titles. We run the algorithms on each task for at least 15 independent trials. The curves show the average simple regret after standardization, while the shaded area denotes the 95\% confidence interval through the optimization.
}
\label{fig:exps:all_res}

\end{figure*}

In this section, we empirically study the performance of \algname against \revise{three} baselines, including (1) cEI, the extension of EI into CBO from \citet{gelbart2014bayesian}, (2) cMES-IBO, a state-of-the-art information-based approach by \citet{takeno2022sequential}, and (3) SCBO, a recent Thompson Sampling (TS) method tailored for scalable CBO from \citet{eriksson2021scalable}. We abstain from comparison against Augmented-Lagrangian methods, following the practice of \citet{takeno2022sequential}, as past studies have illustrated its inferior performance against sampling methods \citep{eriksson2021scalable} or information-based methods \citep{takeno2022sequential, hernandez2014predictive}. \revise{We defer the comparison against CONFIG \cite{xu2023constrained} to \appref{sec:config}, due to the difference in objective and a resulting instability on our benchmarks.} We begin by describing the optimization tasks, and then discuss the performances. To guarantee a fair comparison across all methods, results are averaged over multiple independent trials. For each trial, the random seed is set universally based only on the trial number, ensuring that every algorithm is evaluated under identical stochastic conditions. 




\subsection{CBO Tasks}
We compare \algname against the aforementioned baselines across six CBO tasks. The first two synthetic CBO tasks are constructed from conventional BO benchmark tasks \citep{balandat2020botorch}. %\footnote{Here, we rely on the implementation contained in BoTorch's \citep{balandat2020botorch} test function module.}. 
Among the other four real-world CBO tasks, the first three are extracted from \citet{tanabe2020easy}, offering a broad selection of multi-objective multi-constraints optimization tasks. The fourth one is a 32-dimensional optimization task extracted from the UCI Machine Learning repository \citep{misc_wave_energy_converters_534}. Further details about the datasets are available in \appref{sec:dataset}.

\begin{itemize}[leftmargin=*]
  \item The \emph{Rastrigin function} is a non-convex function used as a performance test problem for optimization algorithms. It was first proposed by \citet{10018403158} and used as a popular benchmark dataset \citep{pohlheimgeatbx}. The feasible region takes up approximately 60\% of the search space, which we construct by sampling $\vert \discreteSet\vert=20000$ and reuse for all 15 trials. We also vary the threshold to control the portion of the feasible region to study the robustness of \algname. \Figref{fig:exps:scan_res} shows the distribution of the objective function and feasible regions.
  \item The \emph{Ackley function} is another commonly used optimization benchmark. We construct two constraints to enforce a feasible area of 14\% of the search space, which we construct by sampling $\vert \discreteSet\vert=20000$ and reuse for all 15 trials. \revise{We also include another experiment on continuous search space of Ackley-10D-2C from \cite{eriksson2021scalable} in \appref{sec:boundary} addressing boundary optimum.}
  \item The \emph{pressure vessel design problem} aims at optimizing the total cost of a cylindrical pressure vessel. The feasible regions take up around 78\% of the whole search space. 
  \item The \emph{coil compression spring design problem} aims to optimize the volume of spring steel wire, which is used to manufacture the spring \citep{lampinen1999mixed} under static loading. The feasible regions take up approximately 0.38\% of the whole search space.
  \item The \emph{car cab design problem} includes seven input variables and eight constraints. The feasible region takes up approximately 13\% of the whole search space.
  \item This \emph{UCI water converter problem} consists of positions and absorbed power outputs of wave energy converters (WECs) from the southern coast of Sydney\citep{misc_wave_energy_converters_534}. The feasible region takes up approximately 27\% of the whole search space.
\end{itemize}


\subsection{Results}
We study the robustness of the algorithms with varying feasible region sizes on the Rastrigin-1D-1C task. Results are demonstrated in \figref{fig:exps:scan_res}. 
Note that the discrete search space consists of the 1000 points shown in the first row of \figref{fig:exps:scan_res}, and with the observation noises, only \algname consistently reaches the global optimum within 2000 iterations. The convergence highlights the essential role of the active learning of the constraint in achieving robust optimization when unknown constraints are present. 

We further study \algname on the aforementioned optimization tasks, with simple regret curves shown in \figref{fig:exps:all_res}. On the Rastrigin-1D-1C and Car-Cabin-7D-8C tasks, \algname initially lags behind the baselines. This is likely because the computational demands of actively learning the constraints temporarily hinder optimization progress. However, the steady improvement of \algname leads to consistently superior performance after sufficient iterations, whereas the baselines become trapped in local optima. These results demonstrate that \algname is efficient and effective across various input dimensionalities, constraint numbers, and constraint correlations. For further details, \appref{sec:additional_exps} includes \Tabref{table:experiments-SVM-CBO}, which presents the mean simple regret at specific budget points, and \Tabref{tab:average_rank}, which summarizes performance ranks within 100 iterations. A comparison of wall-clock times is provided in \tabref{table: walltime}.

