SVM why to switch to dual Lagrange?

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I am newbie reading Support Vector Machines Explained. Translation to dual Quadratic program (p.3 - p.4) in SVM explaining article I mentioned confused me because I understand basic Lagrange multiplier approach but do not understand duality.

My question is:

Could

$\frac{1}{2} {\parallel\omega\parallel}^{2} - \sum_{i=1}^{L}{\alpha}_{i}[{y}_{i}({x}_{i}\omega - b)-1]$

be solved directly without being translated to dual Lagrange task?