Optimal parameters

Question

Using lagrangian, we could get optimal parameters of linear hard margin SVM which are $w^*$ and $b^*$. How can we approach this problem of proving that these parameters are indeed optimal?

Answer 1

Hard margin SVM is a convex problem. Both the objective function and constraints are convex. You can use KKT condition to prove that $w^*$ and $b^*$ are indeed optimal. Just check if the Lagrange multipliers are positive(usual notation).
Notation: $$\ f(x)\leq0\\ g_i(x)\leq0 \ \forall i \in [n]\\ \mathcal L(x, \lambda_{[n]}) = f(x)+\sum_{i=1}^n\lambda_ig_i(x) $$ KKT Condition: Suppose you want to verify if $x^*$ is the optimal,
If there exists $\lambda_i\geq0 \ \forall i \in [n]$ such that, $$\ \nabla f(x^*) + \sum_{i=1}^n\lambda_i\nabla g_i(x^*)=0 $$ Then, $x^*$ is indeed the optimal. There are some regulatory conditions but you need not worry about them as constraints for SVM are all linear. Note that you can directly use the Lagrange multipliers obtained.