Summary: This article gives a comprehensive study on SMO-type (Sequential Minimal Optimization) decomposition methods for training support vector machines. We propose a general and flexible selection of the two-element working set. Main theoretical results include 1) a simple asymptotic convergence proof, 2) a useful explanation of the shrinking and caching techniques, and 3) the linear convergence of this method. This analysis applies to any SMO-type implementation whose selection falls into the proposed framework.