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I was reading the A Tutorial on Support Vector Machines for Pattern Recognition as a supplemental for my Intro to ML class and I wasn't sure why $ a_i \geq 0 $ and $ \lambda_i \geq 0 $ cannot be negative in this case. Can they be negative for any other similar types of optimization problems?

$$ \begin{array}{c} \min \frac{1}{2} ||w||^2 + \frac{C}{n}\sum_i \xi_i + \sum_i \alpha_i (1-y_i w^T x_i - \xi_i) - \sum_i \lambda_i \xi_i \\ \alpha_i \geq 0, \xi_i \geq 0, \lambda_i \geq 0 \end{array} $$

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This follows directly since this is the Langragian dual - dual variables must be non-negative.

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