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I would like to maximize the minimum spacing between a fixed number of points ($x_i \in \mathbb{R}^2$) placed inside a polygon in the plane. The minimum spacing includes distance to the polygon.

Thus, I am trying to solve $$ \max_{x_i,R} R \\ s.t. \quad ||x_i -x_j||_2 \geq R, \quad \forall\ i \neq j \\ a_k^Tx_i + R||a_k||_2 \leq b_k, \quad \forall\: i,k\\ Ax_i \leq b, \quad \forall \: i $$

However, the first set of constraints are not convex. Is there any known relaxation or approximation for this problem?

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This is a fairly complex variant of the circle packing problem (just google it): in that case you are given $n$ circles of the same radius $R$ and you want to determine the maximal $R$ and their center position so that they do not overlap and stay in a unitary square.

Is is a non convex quite hard problem. Yours sounds even harder. There are some convex relaxations but usually quite loose. For instance you might look to a series of paper from KM Anstreicher about SDP relaxations.

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    $\begingroup$ Agreed. I've actually tried models like this using convex SDP-based relaxations. They don't work. :-) $\endgroup$ – Michael Grant Mar 20 '14 at 20:10
  • $\begingroup$ Ah, for some reason I thought the circle-packing problem was maximizing $n$ given fixed $R$ and that this would be somewhat different, but I guess not. $\endgroup$ – Ross B. Mar 22 '14 at 18:43

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