SVM with gradient descent

The constrained optimization problem in SVM is given by
min 1/2 ||w||^2
s.t y(i)(w^T x(i) + b >= 1 for all i

Now converting this to an unconstrained optimization problem gives the lagriangian L as shown in the picture

And while deriving this we also get an equation for w as shown in picture
I’ve been instructed to implement this using gradient descent and give reason if this is not possible
This is the approach I tried to follow:

1. Use gradient descent to minimize the lagrangian and get the minimum alpha vector
2. Then using equation for w which uses this alpha vector …compute w vector
   Also I’ve included b (intercept) in w itself .

But it’s not working
I’m very new to ML and I couldn’t understand all the math behind this. Help