Why do derivations for SVM not consider slack variables for inequality constraints?

(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.

If you introduce the slack variables, then we do not impose a sign constraint on $$\alpha_i$$ but we impose a sign constraint on $$s_i \ge 0$$.
If we do not introduce the slack variables (which is the common practice). We ends up with the sign constraint that $$\alpha_i \ge 0$$.
• This raises more follow up questions in my head - (1) Does that mean that the constraint $\alpha >= 0$ doesn't hold for Lagrangians with equality constraints? I thought the $\alpha$'s were just proportionality constants that could be positive or negative? (2) Let's call Lagrange multipliers used when we introduce slack variables $\lambda$ and when we don't introduce slack variables $\alpha$, are you suggesting $\alpha 𝑦_𝑖(𝑤^𝑇𝑥_i+𝑏-1) = \lambda 𝑦_𝑖(𝑤^𝑇𝑥_i+𝑏-1-s_i)$ where $\alpha>=0$ and $s_i>=0$ Mar 3, 2019 at 7:16
• If you introduce slack variable to make the inequality constraint becoming an equality constraint, and then we take the dual, there is no sign constraint on $\alpha_i$. At the optimal solution, from the complementary slackness condition we do have $\alpha_i y_i(w^Tx_i+b-1) = 0=\lambda_i y_i(w^Tx_i + b -1-s_i)$. Mar 3, 2019 at 7:49