Convergence of decomposition methods for support vector machines

Decomposition methods play an important role in solving large-scale quadratic programming (QP) problems arising from support vector machines (SVMs). In this paper, we study convergence of general decomposition methods for SVMs. We prove that, under a mild condition on the working set selection, a decomposition algorithm stops within a finite number of iterations after reaching a solution of the QP problem satisfying a relaxed Karush-Kuhn-Tucker (KKT) condition which has been often used so far. Further, it is shown that the working set selection used in the implementation of SVMlight satisfies the condition given in this paper, so SVMlight has the finite termination property without the stronger assumption than the positive-semi-definiteness on the Hessian matrix of the objection function.