v2007.09.17 - Convex Optimization

6.9. THEOREM OF THE ALTERNATIVE 435To find the dual EDM cone in ambient S N h per2.13.9.4 we prune theaggregate in (1054) describing the ordinary dual EDM cone, removing anymember having nonzero main diagonal:EDM N∗ ∩ S N h = cone { δ 2 (V N υυ T V T N ) − V N υυT V T N | υ ∈ RN−1}= {δ 2 (V N ΨV T N ) − V N ΨV T N| Ψ∈ SN−1 + }(1102)When N = 1, the EDM cone and its dual in ambient S h each comprisethe origin in isomorphic R 0 ; thus, self-dual in this dimension. (confer (84))When N = 2, the EDM cone is the nonnegative real line in isomorphic R .(Figure 103) EDM 2∗ in S 2 h is identical, thus self-dual in this dimension.This [ result]is in agreement [ with ](1100), verified directly: for all κ∈ R ,11z = κ and δ(zz−1T ) = κ 2 ⇒ d ∗ 12 ≥ 0.1The first case adverse to self-duality N = 3 may be deduced fromFigure 95; the EDM cone is a circular cone in isomorphic R 3 corresponding tono rotation of the Lorentz cone (147) (the self-dual circular cone). Figure 108illustrates the EDM cone and its dual in ambient S 3 h ; no longer self-dual.6.9 Theorem of the alternativeIn2.13.2.1.1 we showed how alternative systems of generalized inequalitycan be derived from closed convex cones and their duals. This section is,therefore, a fitting postscript to the discussion of the dual EDM cone.6.9.0.0.1 Theorem. EDM alternative. [113,1]Given D ∈ S N hD ∈ EDM Nor in the alternative{1 T z = 1∃ z such thatDz = 0(1103)In words, either N(D) intersects hyperplane {z | 1 T z =1} or D is an EDM;the alternatives are incompatible.⋄