This paper proposes CONFIG, a constrained efficient global optimization algorithm for expensive black-box functions. The algorithm uses lower confidence bound (LCB) surrogates to select the next sample, ensuring both objective and constraint feasibility. Under regularity assumptions, CONFIG achieves the same cumulative regret bound as the unconstrained case and similar cumulative constraint violation bounds. For commonly used kernels like Mátern and Squared Exponential, the bounds are sublinear, allowing derivation of convergence rates to the optimal solution. CONFIG naturally detects infeasibility when the original problem is infeasible. Numerical experiments on sampled Gaussian process instances, artificial problems, and a building controller tuning task show that CONFIG performs competitively with state-of-the-art methods, achieving better theoretical guarantees and empirical performance. The algorithm is simple, efficient, and robust to infeasibility, making it suitable for real-world applications with expensive black-box functions and constraints.This paper proposes CONFIG, a constrained efficient global optimization algorithm for expensive black-box functions. The algorithm uses lower confidence bound (LCB) surrogates to select the next sample, ensuring both objective and constraint feasibility. Under regularity assumptions, CONFIG achieves the same cumulative regret bound as the unconstrained case and similar cumulative constraint violation bounds. For commonly used kernels like Mátern and Squared Exponential, the bounds are sublinear, allowing derivation of convergence rates to the optimal solution. CONFIG naturally detects infeasibility when the original problem is infeasible. Numerical experiments on sampled Gaussian process instances, artificial problems, and a building controller tuning task show that CONFIG performs competitively with state-of-the-art methods, achieving better theoretical guarantees and empirical performance. The algorithm is simple, efficient, and robust to infeasibility, making it suitable for real-world applications with expensive black-box functions and constraints.