How would I go about minimizing the expression
$\left(|z_1| + |z_2|\right) \times \left(|z_1 + z_2| + |z_1 - z_2|\right)$
subject to the constraint
$|z_1|^2 + |z_2|^2 = 1$
given that $z_1$ and $z_2$ can be complex numbers?
I thought of trying Lagrange multipliers, but it doesn't seem possible because there are an infinite number of solutions (and solving a 5-equation, 5-variable system is a bit painful).
Any hints on how I could do this?