<p>This article proposes a novel hybrid Bayesian optimization framework designed to solve problems with equality, inequality constraints, and their combinations. We develop a new variant of the expected improvement acquisition function that effectively addresses constraints during iteration. This function balances exploration and exploitation within the constrained search space. Our framework adeptly manages inequality and equality constraints and can initiate the iterative process even from infeasible starting points. Finally, we evaluate our proposed framework against existing approaches using synthetic test problems and a real-world engineering design and hydrology problem. By integrating Bayesian optimization techniques with advanced constraint handling, our framework provides a promising avenue for addressing mixed constraints in black-box optimization scenarios.</p>