SVM and support vector

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I had gone through several example of support vector machines (SVM) and I see one starts explaining SVM by picking up the support vectors upfront (like this). Basically, those vectors that fall on the decision boundaries are picked up upfront. As I was new to SVM, I was wondering how would a machine do this. What is the mechanism used, is the Euclidean distance between two points (vectors) the key?

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Bacically, SVM amounts to having a computer numerically solve an optimization problem. SVM, in its standard form, boils down to a quadratic programming (QP) problem, which has many desirable features which make it easy to solve including convexity and a linear gradient. There are several codes out there to solve SVMs such as libsvm which you can download and use.