Self-scaled barriers and interior-point methods for convex programming
Mathematics of Operations Research
Mathematical Programming: Series A and B
A Riemannian Framework for Tensor Computing
International Journal of Computer Vision
Largest dual ellipsoids inscribed in dual cones
Mathematical Programming: Series A and B
Primal Central Paths and Riemannian Distances for Convex Sets
Foundations of Computational Mathematics
Journal of Mathematical Imaging and Vision
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In this paper we show that the radius of the largest symmetric gauge ball inscribed in the convex cone ${\mathrm{PD}}(n)$ of $n\times n$ positive definite matrices equipped with the Finsler metric inherited from a unitarily invariant norm is precisely the minimal eigenvalue of its center. We then solve Todd's largest dual ellipsoids problem [Math. Program. Ser. B, 117 (2009), pp. 425-434] for unitarily invariant norms, i.e., the problem of maximizing the product of the unitarily invariant norm distances to boundaries of the cone and its dual cone. We further show that the optimal set of maximizers forms a convex cone and is a closed geodesically convex subset of the Riemannian manifold ${\mathrm{PD}}(n)$. This in particular provides a one-parameter family of strictly increasing solid convex (in both a Euclidean and a Riemannian sense) cones of positive definite matrices which starts from the optimal set and covers ${\mathrm{PD}}(n)$.