v2009.01.01 - Convex Optimization

154 CHAPTER 2. CONVEX GEOMETRY
2.13.2.1 Null certificate, Theorem of the alternative
If in particular x p /∈ K a closed convex cone, then the construction
in Figure 48(b) suggests there exists a supporting hyperplane (having
inward-normal belonging to dual cone K ∗ ) separating x p from K ; indeed,
(288)
x p /∈ K ⇔ ∃ y ∈ K ∗ 〈y , x p 〉 < 0 (303)
The existence of any one such y is a certificate of null membership. From a
different perspective,
x p ∈ K
or in the alternative
∃ y ∈ K ∗ 〈y , x p 〉 < 0
(304)
By alternative is meant: these two systems are incompatible; one system is
feasible while the other is not.
2.13.2.1.1 Example. Theorem of the alternative for linear inequality.
Myriad alternative systems of linear inequality can be explained in terms of
pointed closed convex cones and their duals.
Beginning from the simplest Cartesian dual generalized inequalities (289)
(with respect to the nonnegative orthant R m + ),
y ≽ 0 ⇔ x T y ≥ 0 for all x ≽ 0 (305)
Given A∈ R n×m , we make vector substitution y ← A T y
A T y ≽ 0 ⇔ x T A T y ≥ 0 for all x ≽ 0 (306)
Introducing a new vector by calculating b ∆ = Ax we get
A T y ≽ 0 ⇔ b T y ≥ 0, b = Ax for all x ≽ 0 (307)
By complementing sense of the scalar inequality:
A T y ≽ 0
or in the alternative
b T y < 0, ∃ b = Ax, x ≽ 0
(308)