v2009.01.01 - Convex Optimization

6.9. THEOREM OF THE ALTERNATIVE 491
To find the dual EDM cone in ambient S N h per2.13.9.4 we prune the
aggregate in (1153) describing the ordinary dual EDM cone, removing any
member 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 + }
(1201)
When N = 1, the EDM cone and its dual in ambient S h each comprise
the origin in isomorphic R 0 ; thus, self-dual in this dimension. (confer (95))
When N = 2, the EDM cone is the nonnegative real line in isomorphic R .
(Figure 122) EDM 2∗ in S 2 h is identical, thus self-dual in this dimension.
This [ result]
is in agreement [ with ](1199), verified directly: for all κ∈ R ,
1
1
z = κ and δ(zz
−1
T ) = κ 2 ⇒ d ∗ 12 ≥ 0.
1
The first case adverse to self-duality N = 3 may be deduced from
Figure 114; the EDM cone is a circular cone in isomorphic R 3 corresponding
to no rotation of the Lorentz cone (160) (the self-dual circular cone).
Figure 127 illustrates the EDM cone and its dual in ambient S 3 h ; no longer
self-dual.
6.9 Theorem of the alternative
In2.13.2.1.1 we showed how alternative systems of generalized inequality
can 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. [137,1]
Given D ∈ S N h
D ∈ EDM N
or in the alternative
{
1 T z = 1
∃ z such that
Dz = 0
(1202)
In words, either N(D) intersects hyperplane {z | 1 T z =1} or D is an EDM;
the alternatives are incompatible.
⋄