  We study the mixed-integer optimization (MIO) approach to feature subset
selection in nonlinear kernel support vector machines (SVMs) for binary
classification. First proposed for linear regression in the 1970s, this
approach has recently moved into the spotlight with advances in optimization
algorithms and computer hardware. The goal of this paper is to establish an MIO
approach for selecting the best subset of features for kernel SVM
classification. To measure the performance of subset selection, we use the
kernel-target alignment, which is the distance between the centroids of two
response classes in a high-dimensional feature space. We propose a
mixed-integer linear optimization (MILO) formulation based on the kernel-target
alignment for feature subset selection, and this MILO problem can be solved to
optimality using optimization software. We also derive a reduced version of the
MILO problem to accelerate our MILO computations. Experimental results show
good computational efficiency for our MILO formulation with the reduced
problem. Moreover, our method can often outperform the linear-SVM-based MILO
formulation and recursive feature elimination in prediction performance,
especially when there are relatively few data instances.
