Considering these 3 equations:
$$ \alpha_{1} = \arg \max\limits_{\alpha \geq 0} \quad \Big \{ (\alpha - 1) \big (\frac{1/\alpha}{1/\alpha + 1/2 \alpha_{2} + 1/2 \alpha_{3}} \big)^2 \Big \} \\ \alpha_{2} = \arg \max \limits_{\alpha \geq 0} \quad \Big \{ (\alpha - 1) \big (\frac{1/2 \alpha}{1/\alpha_{1} + 1/2 \alpha + 1/2 \alpha_{3}} \big)^2 \Big \} \\ \alpha_{3} = \arg \max \limits_{\alpha \geq 0} \quad \Big \{ (\alpha - 1) \big (\frac{1/2 \alpha}{1/\alpha_{1} + 1/2 \alpha_{2} + 1/2 \alpha} \big)^2 \Big \} \\ $$
What would be the most appropriate numerical method to solve the system ? Thanks.
