The algebraic degree of semidefinite programming

  • Authors:

  • Jiawang Nie;Kristian Ranestad;Bernd Sturmfels

  • Affiliations:

  • UC San Diego, Department of Mathematics, 92093, La Jolla, CA, USA;University of Oslo, Department of Mathematics, PB 1053, Blindern, 0316, Oslo, Norway;UC Berkeley, Department of Mathematics, 94720, Berkeley, CA, USA

  • Venue:
  • Mathematical Programming: Series A and B
  • Year:
  • 2009

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Abstract

Given a generic semidefinite program, specified by matrices with rational entries, each coordinate of its optimal solution is an algebraic number. We study the degree of the minimal polynomials of these algebraic numbers. Geometrically, this degree counts the critical points attained by a linear functional on a fixed rank locus in a linear space of symmetric matrices. We determine this degree using methods from complex algebraic geometry, such as projective duality, determinantal varieties, and their Chern classes.