  Bayesian optimization (BO) has proven to be an effective paradigm for the
global optimization of expensive-to-sample systems. One of the main advantages
of BO is its use of Gaussian processes (GPs) to characterize model uncertainty
which can be leveraged to guide the learning and search process. However, BO
typically treats systems as black-boxes and this limits the ability to exploit
structural knowledge (e.g., physics and sparse interconnections). Composite
functions of the form $f(x, y(x))$, wherein GP modeling is shifted from the
performance function $f$ to an intermediate function $y$, offer an avenue for
exploiting structural knowledge. However, the use of composite functions in a
BO framework is complicated by the need to generate a probability density for
$f$ from the Gaussian density of $y$ calculated by the GP (e.g., when $f$ is
nonlinear it is not possible to obtain a closed-form expression). Previous work
has handled this issue using sampling techniques; these are easy to implement
and flexible but are computationally intensive. In this work, we introduce a
new paradigm which allows for the efficient use of composite functions in BO;
this uses adaptive linearizations of $f$ to obtain closed-form expressions for
the statistical moments of the composite function. We show that this simple
approach (which we call BOIS) enables the exploitation of structural knowledge,
such as that arising in interconnected systems as well as systems that embed
multiple GP models and combinations of physics and GP models. Using a chemical
process optimization case study, we benchmark the effectiveness of BOIS against
standard BO and sampling approaches. Our results indicate that BOIS achieves
performance gains and accurately captures the statistics of composite
functions.
