Simple proof of convergence of the SMO algorithm for different SVM variants

In this brief, we give a new proof of the asymptotic convergence of the sequential minimum optimization (SMO) algorithm for both the most violating pair and second order rules to select the pair of coefficients to be updated. The proof is more self-contained, shorter, and simpler than previous ones and has a different flavor, partially building upon Gilbert's original convergence proof of its algorithm to solve the minimum norm problem for convex hulls. It is valid for both support vector classification (SVC) and support vector regression, which are formulated under a general problem that encompasses them. Moreover, this general problem can be further extended to also cover other support vector machines (SVM)-related problems such as -SVC or one-class SVMs, while the convergence proof of the slight variant of SMO needed for them remains basically unchanged.