\begin{abstract}
Optimizing objectives under constraints, where both the objectives and constraints are black box functions, is a common challenge in real-world applications such as medical therapy design, industrial process optimization, and hyperparameter optimization. Bayesian Optimization (BO) is a popular approach for tackling these complex scenarios. 
However, constrained Bayesian Optimization (CBO) often relies on heuristics, approximations, or relaxation of objectives, which can lead to weaker theoretical guarantees compared to canonical BO. In this paper, we address this gap by focusing on identifying the interior optimum of the constrained objective, deliberately excluding boundary candidates susceptible to noise perturbations. 
Our approach leverages the insight that jointly optimizing the objective and learning the constraints can help pinpoint high-confidence \emph{regions of interest} (ROI) likely to contain the interior optimum. We introduce an efficient CBO framework, which intersects these ROIs within a discretized search space to determine a general ROI. Within this ROI, we optimize the acquisition functions, balancing constraints learning and objective optimization. We showcase the efficiency and robustness of our proposed framework by deriving high-probability regret bounds and validating its performance through extensive empirical evaluations.
\end{abstract}


