How to solve alphas (or the dual equation) after getting Lagrangian dual of SVM

Question

I'm trying to learn SVM by myself, and I'm stuck after getting the dual of SVM. I understand getting the dual after the primal. But, I am stuck here. Please help.

We assume that the hard margin case here (the data of two classes are perfectly linearly separable).

$$ MAX ∑𝛼_𝑖−1/2∑∑𝑦_𝑖𝑦_𝑗𝛼_𝑖𝛼_𝑗(𝑥_𝑖)^𝑇𝑥_𝑗 \\ or \\ MIN 1/2∑∑𝑦_𝑖𝑦_𝑗𝛼_𝑖𝛼_𝑗(𝑥_𝑖)^𝑇𝑥_𝑗 - ∑𝛼_𝑖 $$

1)How do we solve that equation from here? In other words, how do we get alphas? I've searched online and watched youtube, but no places really explain what happens after this equation. Most of them just move to the idea of either soft-margin case, the Kernel tricks, or "use quadratic programming or KKT to solve it". Could anyone show me the steps needed to solve alphas after this?

2)So, intuitively do we get what are the support vectors first? Or, Do we get alpha values first, and then compute those to find out which are actually support vectors?

Please help. Thank you so much in advance. And, this is my first time asking a question here, so forgive me with the notation error no i's and j's on Sums etc.

Answer 1

I am not going to give the full answer on how to compute the solution to the dual equation, i.e. $\alpha$* but I will give you the names of the techniques you can use to obtain $\alpha$*:

• Projected Gradient Ascent
• Sequential Minimal Optimization

As for the second question you can get the optimal $w$* from $\alpha$* using the formula:

$$w^{*} = X^{T}Y \alpha^{*}$$

where X is the dataset and Y is the vector of corresponding class labels.