v2009.01.01 - Convex Optimization

152 CHAPTER 2. CONVEX GEOMETRY
2.13.2 Abstractions of Farkas’ lemma
2.13.2.0.1 Corollary. Generalized inequality and membership relation.
[173,A.4.2] Let K be any closed convex cone and K ∗ its dual, and let x
and y belong to a vector space R n . Then
y ∈ K ∗ ⇔ 〈y , x〉 ≥ 0 for all x ∈ K (287)
which is, merely, a statement of fact by definition of dual cone (270). By
closure we have conjugation:
x ∈ K ⇔ 〈y , x〉 ≥ 0 for all y ∈ K ∗ (288)
which may be regarded as a simple translation of the Farkas lemma [111]
as in [266,22] to the language of convex cones, and a generalization of the
well-known Cartesian fact
x ≽ 0 ⇔ 〈y , x〉 ≥ 0 for all y ≽ 0 (289)
for which implicitly K = K ∗ = R n + the nonnegative orthant.
Membership relation (288) is often written instead as dual generalized
inequalities, when K and K ∗ are pointed closed convex cones,
x ≽
K
0 ⇔ 〈y , x〉 ≥ 0 for all y ≽ 0 (290)
K ∗
meaning, coordinates for biorthogonal expansion of x (2.13.8) [318] must
be nonnegative when x belongs to K . By conjugation [266, thm.14.1]
y ≽
K ∗
0 ⇔ 〈y , x〉 ≥ 0 for all x ≽
K
0 (291)
⋄
When pointed closed convex cone K is not polyhedral, coordinate axes
for biorthogonal expansion asserted by the corollary are taken from extreme
directions of K ; expansion is assured by Carathéodory’s theorem (E.6.4.1.1).
We presume, throughout, the obvious:
x ∈ K ⇔ 〈y , x〉 ≥ 0 for all y ∈ K ∗ (288)
⇔
x ∈ K ⇔ 〈y , x〉 ≥ 0 for all y ∈ K ∗ , ‖y‖= 1
(292)