The standard form for convex optimisation is usually depicted in the following manner $$ \begin{aligned} \text{minimize} \; &f_0(x) \\ \text{subject to}\; &f_i(x) \leq 0, \; i=1,...m \\ &h_i(x) = 0 \; i=1,...p \end{aligned} $$ In Stephen Boyd's convex optimisation book, the Lagrangian is then expressed as $L(x, \lambda, \nu)$. The whole chapter on duality utilises this standard form of the primal problem.
The SVM hard margin problem is defined as $$ \begin{aligned} \text{minimize}& _{w,b} \frac{1}{2}||w||^2 \\ \text{subject to}& \;\;y^{(i)}(w^Tx^{(i)} + b) \geq 1 \end{aligned} $$
From this, I can see that $f_0(x)$ now becomes $f_0(w)$, but now, $f_i(x)$ is not in terms of $f_i(w,b)$. How do we reconcile this difference and find the Lagrangian with respect to this problem.