  A central problem of materials science is to determine whether a hypothetical
material is stable without being synthesized, which is mathematically
equivalent to a global optimization problem on a highly non-linear and
multi-modal potential energy surface (PES). This optimization problem poses
multiple outstanding challenges, including the exceedingly high dimensionality
of the PES and that PES must be constructed from a reliable, sophisticated,
parameters-free, and thus, very expensive computational method, for which
density functional theory (DFT) is an example. DFT is a quantum mechanics based
method that can predict, among other things, the total potential energy of a
given configuration of atoms. DFT, while accurate, is computationally
expensive. In this work, we propose a novel expansion-exploration-exploitation
framework to find the global minimum of the PES. Starting from a few atomic
configurations, this ``known'' space is expanded to construct a big candidate
set. The expansion begins in a non-adaptive manner, where new configurations
are added without considering their potential energy. A novel feature of this
step is that it tends to generate a space-filling design without the knowledge
of the boundaries of the domain space. If needed, the non-adaptive expansion of
the space of configurations is followed by adaptive expansion, where
``promising regions'' of the domain space (those with low energy
configurations) are further expanded. Once a candidate set of configurations is
obtained, it is simultaneously explored and exploited using Bayesian
optimization to find the global minimum. The methodology is demonstrated using
a problem of finding the most stable crystal structure of Aluminum.
