I want to implement a simple SVM classifier, in the case of high-dimensional binary data (text), for which I think a simple linear SVM is best. The reason for implementing it my
Are you interested in using kernels or not? Without kernels, the best way to solve these kinds of optimization problems is through various forms of stochastic gradient descent. A good version is described in http://ttic.uchicago.edu/~shai/papers/ShalevSiSr07.pdf and that has an explicit algorithm.
The explicit algorithm does not work with kernels but can be modified; however, it would be more complex, both in terms of code and runtime complexity.