Support Vector Machines with soft margin: solving the dual form
if you want to find an analytical solution for the problem I think that it doesn't exist.
At this point you should apply decomposition (don't try to work on all your variables,
but iteratively considers only a subset of them called working set), if (and only if) you use a working set W, |W|=2, then your subproblem has an analytical solution.
Due to the large number of alpha variables, decomposition is the basis of svm implementations, in particular the svm light algorithm uses a working set W of cardinality 2.