Abstract
Suppose x̄ and s̄ lie in the interiors of a cone K and its dual K *, respectively. We seek dual ellipsoidal norms such that the product of the radii of the largest inscribed balls centered at x̄ and s̄ and inscribed in K and K *, respectively, is maximized. Here the balls are defined using the two dual norms. When the cones are symmetric, that is self-dual and homogeneous, the solution arises directly from the Nesterov–Todd primal–dual scaling. This shows a desirable geometric property of this scaling in symmetric cone programming, namely that it induces primal/dual norms that maximize the product of the distances to the boundaries of the cones.
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Dedicated to Steve Robinson on the occasion of his 65th birthday.
This author was supported in part by NSF through grants DMS-0209457 and DMS-0513337 and ONR through grant N00014-02-1-0057.
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Todd, M.J. Largest dual ellipsoids inscribed in dual cones. Math. Program. 117, 425–434 (2009). https://doi.org/10.1007/s10107-007-0171-z
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DOI: https://doi.org/10.1007/s10107-007-0171-z