Multi-class classification is an important and
on-going research subject in machine learning. Recently, the
$\nu$-K-SVCR method was proposed by the authors for multi-class
classification. Since many optimization problems have to be solved
in multi-class classification, it is extremely important to
develop an algorithm that can solve those optimization problems
efficiently. In this paper, the optimization problem in the
$\nu$-K-SVCR method is reformulated as an affine box constrained
variational inequality problem with a positive semi-definite
matrix, and a regularized version of the nonsmooth Newton method
that uses the D-gap function as a merit function is applied to
solve the resulting problems. The proposed algorithm fully
exploits the typical feature of the $\nu$-K-SVCR method, which
enables us to reduce the size of Newton equations significantly.
This indicates that the algorithm can be implemented efficiently
in practice. The preliminary numerical experiments on benchmark
datasets show that the proposed method is considerably faster
than the standard Matlab routine used in the original $\nu$-K-SVCR method.