  Optimizing an expensive, black-box function $f(\cdot)$ is challenging when
its input space is high-dimensional. Sequential design frameworks first model
$f(\cdot)$ with a surrogate function and then optimize an acquisition function
to determine input settings to evaluate next. Optimization of both $f(\cdot)$
and the acquisition function benefit from effective dimension reduction. Global
variable selection detects and removes input variables that do not affect
$f(\cdot)$ across the input space. Further dimension reduction may be possible
if we consider local variable selection around the current optimum estimate. We
develop a sequential design algorithm called Sequential Optimization in Locally
Important Dimensions (SOLID) that incorporates global and local variable
selection to optimize a continuous, differentiable function. SOLID performs
local variable selection by comparing the surrogate's predictions in a
localized region around the estimated optimum with the $p$ alternative
predictions made by removing each input variable. The search space of the
acquisition function is further restricted to focus only on the variables that
are deemed locally active, leading to greater emphasis on refining the
surrogate model in locally active dimensions. A simulation study across three
test functions and an application to the Sarcos robot dataset show that SOLID
outperforms conventional approaches.
