# How to model the constraint of a circle placement problem mathematically?

I have a circle with radius $R$ and $N$ points on a 2D plane with known locations $(x_i,y_i), i=1,2,..N$. I want to place the circle such that it maximizes the number of covered points.

Let $d_i=(x_i-x)^2+(y_i-y)^2$ be the square of the distance between the point $i$ and the potential center of the circle $(x,y)$. To ensure point $i$ is within the circle, $d_i$ must satisfy $d_i\leq R^2$. This condition can be formulated by introducing a binary variable $u_i$ such that $u_i=1$ if point $i$ is inside the circle and $u_i=0$ otherwise. Then, we have

$u_i((x_i-x)^2+(y_i-y)^2)\leq R^2$

My question, does the constraint above model my problem correctly?.

• We discourage "please check whether my answer is correct" questions, as only "yes/no" answers are possible, which won't help you or future visitors. See here and here. Can you edit your post to ask about a specific conceptual issue you're uncertain about? As a rule of thumb, a good conceptual question should be useful even to someone who isn't looking at the problem you happen to be working on. If you just need someone to check your work, you might seek out a friend, classmate, or teacher. – D.W. Jun 27 '17 at 21:44
• What prevents you from answering your question yourself? Is there some conceptual understanding you lack, that prevents you from asking? If so, ask about that. Have you tried proving that it does? Have you tried searching for a counterexample? What progress did you make, and where did you get stuck? – D.W. Jun 27 '17 at 21:45
• – Rodrigo de Azevedo Jul 1 '17 at 10:35