Problem: Find two circles with smallest area to cover all the given finite points on Cartesian plane. Each circle should cover at least two points.
For the simpler problem, when we need to find just one circle to cover all the given points, I know we can solve this optimization problem for given points $x$ and some circle with center at $x_c$ and radius $r$:
\begin{array}{ll}\min\limits_{{r:\,||x-x_c||_2\leq r}} r.\end{array}
For the original problem, we need to minimize $r_1^2+r_2^2$, i.e., $||r||_2$ for $r=[r_1 \quad r_2]$. I am confused what should be the constraint, how to write that either $||x-x_{c_1}||_2\leq r_1$ or $||x-x_{c_2}||_2\leq r_2$ should hold and that each circle should cover at least two points?
I will answer my own question, I manage to find an algorithm for the computational purpose. Maybe will be useful for someone.
We can use the following idea to solve the problem:
i) Create a circle from each given point and slowly increase the radius of each circle.
ii) If any two circles intersect, replace them with one new circle. This new circle will have center in the middle of the line connecting centers of the original two circles.
iii) Repeat (ii) until only 2 circles left.
iv) If circle has only one point inside, increase it until it touches any other point.
You can try to code it, it works!