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Gaddum’s test for symmetric cones

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  • Michael Orlitzky

    (Towson University)

Abstract

A real symmetric matrix A is copositive if $$\left\langle {Ax},{x}\right\rangle \ge 0$$ Ax , x ≥ 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 $$\left\langle {L \left( {x}\right) },{x}\right\rangle \ge 0$$ L x , x ≥ 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.

Suggested Citation

  • Michael Orlitzky, 2021. "Gaddum’s test for symmetric cones," Journal of Global Optimization, Springer, vol. 79(4), pages 927-940, April.
    • Handle: RePEc:spr:jglopt:v:79:y:2021:i:4:d:10.1007_s10898-020-00960-6
    DOI: 10.1007/s10898-020-00960-6
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