v2009.01.01 - Convex Optimization

2.13. DUAL CONE & GENERALIZED INEQUALITY 153
2.13.2.0.2 Exercise. Test of dual generalized inequalities.
Test Corollary 2.13.2.0.1 and (292) graphically on the two-dimensional
polyhedral cone and its dual in Figure 49.
When pointed closed convex cone K
(confer2.7.2.2)
x ≽ 0 ⇔ x ∈ K
x ≻ 0 ⇔ x ∈ rel int K
is implicit from context:
(293)
Strict inequality x ≻ 0 means coordinates for biorthogonal expansion of x
must be positive when x belongs to rel int K . Strict membership relations
are useful; e.g., for any proper cone K and its dual K ∗
x ∈ int K ⇔ 〈y , x〉 > 0 for all y ∈ K ∗ , y ≠ 0 (294)
x ∈ K , x ≠ 0 ⇔ 〈y , x〉 > 0 for all y ∈ int K ∗ (295)
By conjugation, we also have the dual relations:
y ∈ int K ∗ ⇔ 〈y , x〉 > 0 for all x ∈ K , x ≠ 0 (296)
y ∈ K ∗ , y ≠ 0 ⇔ 〈y , x〉 > 0 for all x ∈ int K (297)
Boundary-membership relations for proper cones are also useful:
x ∈ ∂K ⇔ ∃ y ≠ 0 〈y , x〉 = 0, y ∈ K ∗ , x ∈ K (298)
y ∈ ∂K ∗ ⇔ ∃ x ≠ 0 〈y , x〉 = 0, x ∈ K , y ∈ K ∗ (299)
2.13.2.0.3 Example. Linear inequality. [292,4]
(confer2.13.5.1.1) Consider a given matrix A and closed convex cone K .
By membership relation we have
This implies
Ay ∈ K ∗ ⇔ x T Ay≥0 ∀x ∈ K
⇔ y T z ≥0 ∀z ∈ {A T x | x ∈ K}
⇔ y ∈ {A T x | x ∈ K} ∗ (300)
{y | Ay ∈ K ∗ } = {A T x | x ∈ K} ∗ (301)
If we regard A as a linear operator, then A T is its adjoint. When K is the
self-dual nonnegative orthant (2.13.5.1), for example, then
{y | Ay ≽ 0} = {A T x | x ≽ 0} ∗ (302)