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Gaddum’s test for symmetric cones

Abstract

A real symmetric matrix A is copositive if \(\left\langle {Ax},{x}\right\rangle \ge 0\) for all x in the nonnegative orthant. Copositive programming gained fame when Burer showed that hard nonconvex problems can be formulated as completely-positive programs. Alas, the power of copositive programming is offset by its difficulty: simple questions like “is this matrix copositive?” have complicated answers. In 1958, Jerry Gaddum proposed a recursive procedure to check if a given matrix is copositive by solving a series of matrix games. It is easy to implement and conceptually simple. Copositivity generalizes to cones other than the nonnegative orthant. If K is a proper cone, then the linear operator L is copositive on K if \(\left\langle {L \left( {x}\right) },{x}\right\rangle \ge 0\) for all x in K. Little is known about these operators in general. We extend Gaddum’s test to self-dual and symmetric cones, thereby deducing criteria for copositivity in those settings.

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Correspondence to Michael Orlitzky.

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Orlitzky, M. Gaddum’s test for symmetric cones. J Glob Optim 79, 927–940 (2021). https://doi.org/10.1007/s10898-020-00960-6

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Keywords

  • Gaddum’s test
  • Copositivity
  • Symmetric cone
  • Linear game
  • Cone programming

Mathematics Subject Classification

  • 90C25
  • 91A05
  • 15B48
  • 90C33
  • 90–08