  The problem of optimizing unknown costly-to-evaluate functions has been
studied for a long time in the context of Bayesian Optimization. Algorithms in
this field aim to find the optimizer of the function by asking only a few
function evaluations at locations carefully selected based on a posterior
model. In this paper, we assume the unknown function is Lipschitz continuous.
Leveraging the Lipschitz property, we propose an algorithm with a distinct
exploration phase followed by an exploitation phase. The exploration phase aims
to select samples that shrink the search space as much as possible. The
exploitation phase then focuses on the reduced search space and selects samples
closest to the optimizer. Considering the Expected Improvement (EI) as a
baseline, we empirically show that the proposed algorithm significantly
outperforms EI.
