$\left(1-{\alpha }_1\right)\ge 0$

$\Rightarrow {\alpha }_1\le 1$

$\left({\alpha }_2+\frac{{\alpha }_1}{2}\right)\ge 0$

$\Rightarrow {\alpha }_2\le \frac{{\alpha }_1}{2}$

$\left(\frac{{\alpha }_1}{2}-{\alpha }_2\right)\ge 0$

$\Rightarrow \frac{{\alpha }_1}{2}+{\alpha }_2\ge 0$

$\Rightarrow \frac{{\alpha }_1}{2}+\frac{{\alpha }_1}{4}\ge 0$

$\Rightarrow {\alpha }_1\ge 0$