v2009.01.01 - Convex Optimization

2.13. DUAL CONE & GENERALIZED INEQUALITY 163
⎧⎡
⎨
K = ⎣
⎩
⎧⎡
⎨
= ⎣
⎩
〈A 1 , X〉
.
〈A m , X〉
⎤ ⎫
⎬
⎦ | X ≽ 0
⎭ ⊆ Rm
⎤ ⎫
svec(A 1 ) T
⎬
. ⎦svec X | X ≽ 0
svec(A m ) T ⎭
∆
= {A svec X | X ≽ 0}
(337)
where A∈ R m×n(n+1)/2 , and where symmetric vectorization svec is defined
in (49). Cone K is indeed convex because, by (157)
A svec X p1 , A svec X p2 ∈ K ⇒ A(ζ svec X p1 +ξ svec X p2 )∈ K for all ζ,ξ ≥ 0
(338)
since a nonnegatively weighted sum of positive semidefinite matrices must be
positive semidefinite. (A.3.1.0.2) Although matrix A is finite-dimensional,
K is generally not a polyhedral cone (unless m equals 1 or 2) simply because
X ∈ S n + . Provided the A j matrices are linearly independent, then
rel int K = int K (339)
meaning, the cone interior is nonempty; implying, the dual cone is pointed
by (280).
If matrix A has no nullspace, on the other hand, then (by results in
2.10.1.1 and Definition 2.2.1.0.1) A svec X is an isomorphism in X between
the positive semidefinite cone and R(A). In that case, convex cone K has
relative interior
rel int K = {A svec X | X ≻ 0} (340)
and boundary
rel∂K = {A svec X | X ≽ 0, X ⊁ 0} (341)
Now consider the (closed convex) dual cone:
K ∗ = {y | 〈z , y〉 ≥ 0 for all z ∈ K} ⊆ R m
= {y | 〈z , y〉 ≥ 0 for all z = A svec X , X ≽ 0}
= {y | 〈A svec X , y〉 ≥ 0 for all X ≽ 0}
= { y | 〈svec X , A T y〉 ≥ 0 for all X ≽ 0 }
= { y | svec −1 (A T y) ≽ 0 } (342)