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Decomposition methods are currently one of the major methods for training support vector machines. They vary mainly according to different working set selections. Existing implementations and analysis usually consider some specific selection rules. This paper studies sequential minimal optimization type decomposition methods under a general and flexible way of choosing the two-element working set. The main results include: 1) a simple asymptotic convergence proof, 2) a general explanation of the shrinking and caching techniques, and 3) the linear convergence of the methods. Extensions to some support vector machine variants are also discussed.