  Recently, several Bayesian optimization (BO) methods have been extended to
the expensive black-box optimization problem with unknown constraints, which is
an important problem that appears frequently in practice. We focus on an
information-theoretic approach called Max-value Entropy Search (MES) whose
superior performance has been repeatedly shown in BO literature. Since existing
MES-based constrained BO is restricted to only one constraint, we first extend
it to multiple constraints, but we found that this approach can cause negative
approximate values for the mutual information, which can result in unreasonable
decisions. In this paper, we employ a different approximation strategy that is
based on a lower bound of the mutual information, and propose a novel
constrained BO method called Constrained Max-value Entropy Search via
Information lower BOund (CMES-IBO). Our approximate mutual information derived
from the lower bound has a simple closed-form that is guaranteed to be
nonnegative, and we show that irrational behavior caused by the negative value
can be avoided. Furthermore, by using conditional mutual information, we extend
our methods to the parallel setting in which multiple queries can be issued
simultaneously. Finally, we demonstrate the effectiveness of our proposed
methods by benchmark functions and real-world applications to materials
science.
