The easiest way to deal with this is to split it into two problems: one obtained by adding the inequality $x \ge 0$ and using the objective function $2x$, and the other by adding the inequality $x \le 0$ and using the objective function $x$. Incidentally, I don't think that you mean the inequality $|\alpha_i| \le 0$. Perhaps you mean to omit the absolute value?

The easiest way to deal with this is to split it into two problems: one obtained by adding the inequality $x \ge 0$ and using the objective function $2x$, and the other by adding the inequality $x \le 0$ and using the objective function $x$. Incidentally, I don't think that you mean the inequality $|\alpha_i| \le 0$. Perhaps you mean to omit the absolute value?

I think that the best way of dealing with this is to solve the dual problem. I'll have to think about the details, but that should get the number of vertices in the polytope down to a manageable level.