(This is related to a question I asked a few days ago)
I've been through a few SVM derivations and the ones I follow are this Caltech lecture and this MIT lecture.
However, with both of them the Lagrangian they try to minimize is:
$$\mathcal{L}(\boldsymbol{w}, b, \boldsymbol{\alpha}) = \frac{1}{2}\|\boldsymbol{w}\|^2 + \sum_i \alpha_i\left[y_i(\boldsymbol{w}^T\boldsymbol{x}+b)-1\right].$$
As opposed to: $$\mathcal{L}(\boldsymbol{w}, b, \boldsymbol{\alpha}) = \frac{1}{2}\|\boldsymbol{w}\|^2 + \sum_i \alpha_i\left[y_i(\boldsymbol{w}^T\boldsymbol{x}+b)-1-s_i\right].$$
Where $s_i$ are the slack variables obtained from the inequalities: $$y_i(w^Tx + b) >= 1$$ Hence $$y_i(w^Tx + b) = 1 + s_i$$
Is it because when the inequality is not inactive all the Lagrange multipliers for it are going to be 0 and so we don't even consider them? If yes, then this bit from the Caltech lecture contradicts this statement
This implies that we will get the $\alpha_i$'s as zero but we shouldn't even be talking about them if we're not even considering the inactive equalities in the Lagrangian right.