Semidefinite descriptions of the convex hull of rotation matrices

J. Saunderson, P. A. Parrilo, A. S. Willsky

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We study the convex hull of SO(n), the set of n × n orthogonal matrices with unit determinant, from the point of view of semidefinite programming. We show that the convex hull of SO(n) is doubly spectrahedral, i.e., both it and its polar have a description as the intersection of a cone of positive semidefinite matrices with an affine subspace. Our spectrahedral representations are explicit and are of minimum size, in the sense that there are no smaller spectrahedral representations of these convex bodies.

Original languageEnglish
Pages (from-to)1314-1343
Number of pages30
JournalSIAM Journal on Optimization
Issue number3
Publication statusPublished - 2015
Externally publishedYes


  • Doubly spectrahedral
  • Orbitope
  • Special orthogonal group

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