  Optimizing objectives under constraints, where both the objectives and
constraints are black box functions, is a common scenario in real-world
applications such as scientific experimental design, design of medical
therapies, and industrial process optimization. One popular approach to
handling these complex scenarios is Bayesian Optimization (BO). In terms of
theoretical behavior, BO is relatively well understood in the unconstrained
setting, where its principles have been well explored and validated. However,
when it comes to constrained Bayesian optimization (CBO), the existing
framework often relies on heuristics or approximations without the same level
of theoretical guarantees.
  In this paper, we delve into the theoretical and practical aspects of
constrained Bayesian optimization, where the objective and constraints can be
independently evaluated and are subject to noise. By recognizing that both the
objective and constraints can help identify high-confidence regions of interest
(ROI), we propose an efficient CBO framework that intersects the ROIs
identified from each aspect to determine the general ROI. The ROI, coupled with
a novel acquisition function that adaptively balances the optimization of the
objective and the identification of feasible regions, enables us to derive
rigorous theoretical justifications for its performance. We showcase the
efficiency and robustness of our proposed CBO framework through empirical
evidence and discuss the fundamental challenge of deriving practical regret
bounds for CBO algorithms.
