We propose a bilevel cross-validation scheme for support vector machine (SVM) model selection based on the construction of the entire regularization path. Since such path is a particular case of the more general proximal trajectory concept from nonsmooth optimization, we propose for its construction an algorithm based on solving a finite number of structured linear programs. Our methodology, differently from other approaches, works directly on the primal form of SVM. Numerical results are presented on binary data sets drawn from literature.