2
$\begingroup$

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?

  • 0
    There is a large literature on this sort of thing. Terms to google include linear programming, nonlinear programming, and convex optimization. The function you're trying to optimize is convex, which probably helps, but it's not analytic, which makes it harder. The constraint is nonlinear and not convex, which makes it harder. The whole problem is symmetric with respect to rotation in the complex plane, so you can reduce it to three degrees of freedom by taking $\Im(z_1)=0$ if you wish. It's also symmetric under interchange of $z_1$ and $z_2$, so I suspect you have $|z_1|=0$, $1/\sqrt{2}$,or 1.2012-01-23
  • 0
    Whoa I didn't know any of those terms -- thanks a lot for pointing that out!2012-01-23

2 Answers 2