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In the context of a binary classification problem, the optimal linearcombination of continuous predictors can be estimated by maximizing anempirical estimate of the area under the receiver operating characteristic(ROC) curve (AUC). For multi-category responses, the optimal predictorcombination can similarly be obtained by maximization of the empiricalhypervolume under the manifold (HUM). This problem is particularly relevant tomedical research, where it may be of interest to diagnose a disease withvarious subtypes or predict a multi-category outcome. Since the empirical HUMis discontinuous, non-differentiable, and possibly multi-modal, solving thismaximization problem requires a global optimization technique. Estimation ofthe optimal coefficient vector using existing global optimization techniques iscomputationally expensive, becoming prohibitive as the number of predictors andthe number of outcome categories increases. We propose an efficientderivative-free black-box optimization technique based on pattern search tosolve this problem. Through extensive simulation studies, we demonstrate thatthe proposed method achieves better performance compared to existing methodsincluding the step-down algorithm. Finally, we illustrate the proposed methodto predict swallowing difficulty after radiation therapy for oropharyngealcancer based on radiation dose to various structures in the head and neck.

Estimating the Optimal Linear Combination of Biomarkers using Spherically Constrained Optimization