for a given data set depends on its dimensions as well as its sparsity. Even if $X$ is a sparse matrix, $Q$ in (SVM-D) is still likely to be dense, whereas the Hessian matrix in (SVM-P) is the identity. The primal problem (SVM-P), however, has a larger variable dimension and more constraints. It is often argued that one should solve (SVM-P) when the number of features is smaller than the number of samples, whereas (SVM-D) should be solved when the number of features is less than that of the samples. Since in the second phase of Algorithm \[alg:hipad\] we expect to have identified a small number of promising features, we have decided to solve (SVM-D) by using IPM. Solving (SVM-D) is realized through the OOQP [@gertz2003object] software package that implements a primal-dual IPM for convex QP problems.

The Two-phase Algorithm
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Let us keep in mind that the primary objective for the first phase is to appropriately reduce the feature space dimensionality without impacting the final prediction accuracy. As we mentioned above, the suitability of ADMM for the first phase depends on whether the support of the feature vector converges quickly or not. On an illustrative dataset from [@yeefficient], which 