I have some optimization problem (optimizing parameter $\alpha$)with those constraints: $$\alpha_i\ge0$$ $$\sum\limits_i \alpha_i y_i =0$$ and a third constraints: $$w-\sum\limits_i \alpha_i y_i x_i = 0$$
It was mentioned that the third constraint is unnecessary because any $\alpha$ would satisfy it. I can't immediately see this. Any insight?
It all depends on the values of $x_i$ and $y_i$. In general, the third constraint is most surely not unnecesary. For example, if $w$ is an all zero vector, and $y_1=-1$, $y_2=1$ and $x_1=0, x_2=1$, then the equations are:
$$-\alpha_1 + \alpha_2 = 0\\ -\alpha_2 = 0 $$ which has only $\alpha_1=\alpha_2=0$ as a solution. Without the third constraint, any pair $\alpha,\alpha$ is a solution if $\alpha\geq0$