Soft-Margin SVMs and the SMO Algorithm

The dual derivation assumed the data was perfectly linearly separable — an assumption that fails for essentially all real data, and that even when it holds, makes the classifier dangerously sensitive to a single outlier dragging the whole boundary around. This note relaxes that assumption with soft margins, then closes the topic's optimisation thread by sketching SMO, the coordinate-ascent-style algorithm that actually solves the resulting dual problem in practice (and that underlies sklearn.svm.SVC's solver).