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Suppose we have a set of points $x_1, \ldots, x_N$ in $R^d$ which we call the "bad set." Now we want an ellipse in $R^d$ of maximal volume, centered at point $c$, which does not contain any of the points in the bad set. See the image below, where the red dots are the bad set, and the blue ellipsoid is the sought ellipsoid.

Fig 2.a
Figure 2.a from https://iopscience.iop.org/article/10.1088/1742-6596/1530/1/012087

Are there efficient algorithms known that solve this problem? There are similar problems for which quadratic programming solutions exist, e.g. the minimal volume ellipsoid containing a set of points (Löwner–John), and the maximum volume ellipsoid which lies inside a convex polyhedron. But the only solution I found for this particular problem are heuristic.

Heuristic algorithms

Mark
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  • is the point $c$ given beforehand? – orangeskid Apr 30 '24 at 16:57
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    @orangeskid It is still useful for my application if it must be given beforehand. But I'm certainly interested if an algorithm can place $c$ as well. But note that in any algorithm, some sorts of bounds on $c$ must be given or else one can locate the ellipse infinitely far away from the bad set and grow arbitrarily big. – Mark Apr 30 '24 at 17:01

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