  A new type of experiment that aims to determine the optimal quantities of a
sequence of factors is eliciting considerable attention in medical science,
bioengineering, and many other disciplines. Such studies require the
simultaneous optimization of both quantities and the sequence orders of several
components which are called quantitative-sequence (QS) factors. Given the large
and semi-discrete solution spaces in such experiments, efficiently identifying
optimal or near-optimal solutions by using a small number of experimental
trials is a nontrivial task. To address this challenge, we propose a novel
active learning approach, called QS-learning, to enable effective modeling and
efficient optimization for experiments with QS factors. QS-learning consists of
three parts: a novel mapping-based additive Gaussian process (MaGP) model, an
efficient global optimization scheme (QS-EGO), and a new class of optimal
designs (QS-design). The theoretical properties of the proposed method are
investigated, and optimization techniques using analytical gradients are
developed. The performance of the proposed method is demonstrated via a real
drug experiment on lymphoma treatment and several simulation studies.
