  Deterministic computer simulations are often used as a replacement for
complex physical experiments. Although less expensive than physical
experimentation, computer codes can still be time-consuming to run. An
effective strategy for exploring the response surface of the deterministic
simulator is the use of an approximation to the computer code, such as a
Gaussian process (GP) model, coupled with a sequential sampling strategy for
choosing design points that can be used to build the GP model. The ultimate
goal of such studies is often the estimation of specific features of interest
of the simulator output, such as the maximum, minimum, or a level set
(contour). Before approximating such features with the GP model, sufficient
runs of the computer simulator must be completed.
  Sequential designs with an expected improvement (EI) function can yield good
estimates of the features with a minimal number of runs. The challenge is that
the expected improvement function itself is often multimodal and difficult to
maximize. We develop branch and bound algorithms for efficiently maximizing the
EI function in specific problems, including the simultaneous estimation of a
minimum and a maximum, and in the estimation of a contour. These branch and
bound algorithms outperform other optimization strategies such as genetic
algorithms, and over a number of sequential design steps can lead to
dramatically superior accuracy in estimation of features of interest.
