Is there a solution for problems with non-convex constraints?

Question

Is there a solution to find the global optimum of a convex function , subject to non convex constraints?

For example:

$\min x_1^2 + x_2^2$

subject to

$x_2 + x_1^2 \geq 2$

$x_2 - x_1^2 \leq 3$

constraints

It's only an example; I am looking for a general method to find the optimal point in any convex function with non-convex constraints

Answer 1

The two constraints here are actually both convex, which you can see by drawing out the shape and observing that circles (and lunes) are convex. Alternatively, you know that $x_1^2 + x_2^2$ is a convex function, so $(x_1 - 1)^2 + (x_2 - 1)^2$ is a convex function, and hence the inequality $(x_1 - 1)^2 + (x_2 - 1)^2 \leq 3$ is a convex region.

Also by drawing the regions out, you can see that the origin is included in the feasible set, and $(0, 0)$ clearly minimises $x_1^2 + x_2^2$, so the minimum is $0$.