A D-induced duality and its applications by Brinkhuis J., Zhang S. PDF

By Brinkhuis J., Zhang S.

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A sublinear function φ on U is positive if φ(u) > 0 for all nonzero u. We call two convex cones E ⊆ Y and F ⊆ Z equivalent if there exists a linear bijective transformation Y → Z which maps E bijectively onto F. We say that a convex cone E ⊆ Y is represented by a sublinear mapping φ on U if the convex cones E ⊆ Y and epi(φ) ⊆ × U are equivalent. Now we can formulate and so carry out the second step as follows. 2 Each closed, solid, pointed convex cone C can be represented by a positive sublinear function φ.

This would reduce the problem to that of computing tensor products of cones, in view of the following well-known formula: (Cφ )∗ = Cφ ∗ x¯ ,x where φ ∗ (x) = supx=0 φ(x) for all x ∈ X . We mention in passing that this formula gives an explicit demonstration of the fact that the property of cones to be representable by positive sublinear functions is preserved under taking duals. Unfortunately, the conjectured equality above does not hold in general, although it is obvious that C∗ ⊗ D∗ ⊆ (C ⊗ D)∗ .

SIAM J. Optim. 14, 1140–1162 (2004) 7. : On maximization of quadratic form over intersection of ellipsoids with common center. Math. Progr. 86, 463–473 (1999) 8. : Semidefinite relaxation and nonconvex quadratic optimization. Optim. Methods Software 9, 141–160 (1998) 9. : Global quadratic optimization on the sets with simple structure. Working Paper, Université Catholique de Louvain, Louvain-la-Neuve, Belgium (1999) 10. : Interior point polynomial methods in convex programming. Stud. Appl. Math.

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