v2009.01.01 - Convex Optimization

2.13. DUAL CONE & GENERALIZED INEQUALITY 175
From this, a constrained membership relation for the ordinary dual cone
K ∗ ⊆ R , assuming x,y ∈ M and closed convex cone K ⊆ M
y ∈ K ∗ ∩ M ⇔ 〈y , x〉 ≥ 0 for all x ∈ K (377)
By closure in subspace M we have conjugation (2.13.1.1):
x ∈ K ⇔ 〈y , x〉 ≥ 0 for all y ∈ K ∗ ∩ M (378)
This means membership determination in subspace M requires knowledge
of the dual cone only in M . For sake of completeness, for proper cone K
with respect to subspace M (confer (294))
x ∈ int K ⇔ 〈y , x〉 > 0 for all y ∈ K ∗ ∩ M, y ≠ 0 (379)
x ∈ K , x ≠ 0 ⇔ 〈y , x〉 > 0 for all y ∈ int K ∗ ∩ M (380)
(By conjugation, we also have the dual relations.) Yet when M equals aff K
for K a closed convex cone
x ∈ rel int K ⇔ 〈y , x〉 > 0 for all y ∈ K ∗ ∩ aff K , y ≠ 0 (381)
x ∈ K , x ≠ 0 ⇔ 〈y , x〉 > 0 for all y ∈ rel int(K ∗ ∩ aff K) (382)
2.13.9.4 Subspace M = aff K
Assume now a subspace M that is the affine hull of cone K : Consider again
a pointed polyhedral cone K denoted by its extreme directions arranged
columnar in matrix X such that
rank(X ∈ R n×N ) = N ∆ = dim aff K ≤ n (366)
We want expressions for the convex cone and its dual in subspace M=aff K :
Cone Table A K K ∗ ∩ aff K
vertex-description X X †T
halfspace-description X † , X ⊥T X T , X ⊥T
When dim aff K = n , this table reduces to Cone Table S. These descriptions
facilitate work in a proper subspace. The subspace of symmetric matrices S N ,
for example, often serves as ambient space. 2.69
2.69 The dual cone of positive semidefinite matrices S N∗
+ = S N + remains in S N by convention,
whereas the ordinary dual cone would venture into R N×N .