# Minimize distance given other distance constraints

I want to minimize the Euclidean distance between a pair of points $\mathbf{P}_1, \mathbf{P}_2$ by finding a new point $\mathbf{P}_{2*}$, with the inequality constraints that for each $R_i$ of a set of points $R$, $$\mid\mid \mathbf{P}_{2*}-R_i\mid\mid_2 \leq \mid\mid \mathbf{P}_{2}-R_i\mid\mid_2$$

Geometrically, I could express $\mathbf{R}_i$ as the red points, $\mathbf{P}_2$ as the green point and $\mathbf{P}_1$ as the blue point. Circles with radius of the distance to the green point can then be 'indifference' curves, and help to identify the feasible set.

In the left figure the new point would be the intersection of a line from blue point to red point and the circle - however, in the right figure it would be the intersections of the two red circles.

Since I am not sure if there would be other exceptions, it would be great if someone could perhaps point me to better approaches of solving this. Thank you!

• Are you trying to do this computationally? If so, then the inequality constraints you propose are convex, and the resulting system can be handled with a convex optimization solver. – David M. Jul 22 '18 at 15:08
• Thanks, that was exactly the hint I needed - finally managed to implement it in R and it works great :) @DavidM. – eok Jul 23 '18 at 21:45