# soft SVM - degenerate case

According to "A Note on Support Vector Machine Degeneracy", Theorem 4, if the dual problem for soft-SVM has a solution with $\alpha_i \in \{0,C\}, \forall i$, then $w=0$ for the primal problem.

In "Uniqueness of the SVM solution", there is an example which, I say, contradicts the theorem above:

Data: $x_1 = 1, y_1 = +1; x_2 = -1, y_2 = -1$

$C \in (0,1/2]$

Result: $\alpha_1 = \alpha_2 = C$

$w=2C \neq 0$.

Am I missing something?

Let's check if $$w=2C$$ is actually primal optimal...
The objective value when $$w = 0$$ would be $$2C$$.
When $$w=2C$$ the objective value would be $$\frac{(2C)^2}{2} + (1+b-2C)+(1-b-2C) = 2C^2 - 4C + 2 = 2(C-1)^2$$.
Then $$w=0$$ could be optimal when $$C < (C-1)^2$$ which happens for $$C < \frac{3-\sqrt 5}{2} < \frac{1}{2}$$, but not above it.