By Brinkhuis J., Zhang S.
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Extra info for A D-induced duality and its applications
Math. 13 (1994) 11. : Convex Analysis. Princeton University Press, Princeton (1970) 12. : On cones of nonnegative quadratic functions. Math. Oper. Res. 28, 246– 267 (2003) 13. : Approximating quadratic programming with bound and quadratic constraints. Math. Progr. 84, 219–226 (1999) 14. : A new self-dual embedding method for convex programming. J. Global Optim.
It follows immediately from (23) that t ≥ 0. If t = 0 then (φ(u) + r)bT w + (ψ(w) + s)aT u + uT Mw ≥ 0 for all u ∈ U, w ∈ W, r ≥ 0 and s ≥ 0. This leads to aT u ≥ 0 for all u ∈ U, and bT w ≥ 0 for all w ∈ W. Since U and W are vector spaces, we conclude that a = 0 and b = 0. Therefore the inequality reduces further to uT Mw ≥ 0, ∀u ∈ U and ∀w ∈ W. Since U and W are vector spaces, we get M = 0. Now we consider the situation t > 0. Without losing generality let us scale the value of t and assume t = 1.
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)∗ .