Simple non convex optimization

Question

I am trying to solve the following optimization problem. I'd appreciate any tips or directions.

$ \text{minimize } |x|^2 + |y|^2$

$ \text{subject to } |x-y|^2 \geq 1$

where $|.|$ is the absolute value, and $x$ and $y$ are two complex scalers.

EDITED: The variables are complex

Draw a little picture of the feasible set in the plane. – copper.hat Jun 01 '22 at 15:42 — copper.hat, Jun 01 '22 at 15:42

score 1 · Answer 1 · answered Jun 01 '22 at 15:33

1

Hints

$\lvert x - y \rvert^2 \ge 1$ is equivalent to $\lvert x - y \rvert \ge 1$ and is the union of two disjoint "parallel" half-planes.

$\lvert x \rvert^2 + \lvert y \rvert^2$ is the square of the radius of a circle passing through $(x,y)$ centered on the origin.

answered Jun 01 '22 at 15:33

1 Answers1