  In the context of a binary classification problem, the optimal linear
combination of continuous predictors can be estimated by maximizing an
empirical estimate of the area under the receiver operating characteristic
(ROC) curve (AUC). For multi-category responses, the optimal predictor
combination can similarly be obtained by maximization of the empirical
hypervolume under the manifold (HUM). This problem is particularly relevant to
medical research, where it may be of interest to diagnose a disease with
various subtypes or predict a multi-category outcome. Since the empirical HUM
is discontinuous, non-differentiable, and possibly multi-modal, solving this
maximization problem requires a global optimization technique. Estimation of
the optimal coefficient vector using existing global optimization techniques is
computationally expensive, becoming prohibitive as the number of predictors and
the number of outcome categories increases. We propose an efficient
derivative-free black-box optimization technique based on pattern search to
solve this problem. Through extensive simulation studies, we demonstrate that
the proposed method achieves better performance compared to existing methods
including the step-down algorithm. Finally, we illustrate the proposed method
to predict swallowing difficulty after radiation therapy for oropharyngeal
cancer based on radiation dose to various structures in the head and neck.
