Problems of packing shapes with maximal density, sometimes into a
container of restricted size, are classical in discrete
mathematics. We describe here the problem of packing a given set of
ellipsoids of different sizes into a finite container, in a way that
allows overlap but that minimizes the maximum overlap between adjacent
ellipsoids. We describe a bilevel optimization algorithm for finding
local solutions of this problem, both the general case and the simpler
special case in which the ellipsoids are spheres. Tools from conic
optimization, especially semidefinite programming, are key to the
algorithm. Finally, we describe the motivating application -
chromosome arrangement in cell nuclei - and compare the computational
results obtained with this approach to experimental observations.
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This talk represents joint work with Caroline Uhler (IST Austria).