Why is SVM calculated in this way?

Question

I have a question regarding SVM. I understand the Lagrange equation,

$L(w,b,\alpha) = \frac{1}{2}w'w - \sum_i \alpha_i (y_i(w'x_i+b)-1)=$

$\frac{1}{2}w'w - \sum_i \alpha_i y_i(w'x_i+b)+ \sum_i \alpha_i$

and that taking the derivative to $b$ and $w$ gives

$\frac{\partial L}{\partial b} = \sum_i \alpha_i y_i=0$

$\frac{\partial L}{\partial w} = w - \sum_i \alpha_i y_i x_i=0 \rightarrow w =\sum_i \alpha_i y_i x_i$

Filling this in the Lagrange gives

$L(\alpha)=\frac{1}{2}\sum_i \sum_j \alpha_i \alpha_j y_i y_j x_i' x_j + \sum_i{\alpha_i} $

But this is not how SVM fills in the Lagrange and I like to understand why.

After some time I saw that

$L(w,b,\alpha) = \frac{1}{2}w'w - \sum_i ((w'\alpha_i y_i x_i+\alpha_i y_i b)-\alpha_i y_i)=$

$\frac{1}{2}w'w - \sum_i ((w'w+\alpha_i y_ib)-\alpha_i y_i)$,

which then can be rewritten as the familiar dual problem.

$L(\alpha)=-\frac{1}{2}\sum_i \sum_j \alpha_i \alpha_j y_i y_j x_i' x_j + \sum_i{\alpha_i} $

But what is wrong, or what is the disadvantage of, the first representation?

Answer 1

It seems there's a minus sign missing in the "Filling this in the Lagrange" step .

I got something like $$L = \frac{1}{2}w'w - \sum_i \alpha_i (y_i(w'x_i+b)-1)$$ $$=\frac{1}{2}\sum_i \sum_j \alpha_i \alpha_j y_i y_j x_i' x_j-\sum_i \sum_j \alpha_i \alpha_j y_i y_j x_i' x_j- b\sum_i \alpha_i y_i + \sum_i{\alpha_i} $$ $$=-\frac{1}{2}\sum_i \sum_j \alpha_i \alpha_j y_i y_j x_i' x_j+\sum_i{\alpha_i} $$ which is the same as the other representation.