The Lagrange multipliers in the context of SVMs are typically denoted $\alpha_i$. The fact that one often observes that most $\alpha_i=0$ is a direct consequence of the [Karush-Kuhn-Tucker (KKT)](https://en.wikipedia.org/wiki/Karush%E2%80%93Kuhn%E2%80%93Tucker_conditions) dual complementarity conditions:

[![enter image description here][1]][1]

($y_i(\mathbf{w}^T\mathbf{x}_i+b) = 1$ iff $\mathbf{x}_i$ is on the SVM decision boundary, i.e. is a support vector assuming $\mathbf{x}_i$ is in the training set, and in most cases few training vectors are support vectors, as whuber pointed out in the comments).


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[Andrew Ng's CS229 Lecture notes on SVMs](http://cs229.stanford.edu/notes/cs229-notes3.pdf) introduces the Karush-Kuhn-Tucker (KKT) dual complementarity conditions:

[![enter image description here][2]][2]

[![enter image description here][3]][3]

[![enter image description here][4]][4]

[![enter image description here][5]][5]

Note that we can create some case where all vectors in the training set are support vectors: e.g. see this [Support Vector Machine Question](http://stats.stackexchange.com/q/110598/12359).


  [1]: https://i.sstatic.net/StNDv.png
  [2]: https://i.sstatic.net/7Ch1i.png
  [3]: https://i.sstatic.net/bokzB.png
  [4]: https://i.sstatic.net/IhMHY.png
  [5]: https://i.sstatic.net/qmwvS.png