v2009.01.01 - Convex Optimization

2.13. DUAL CONE & GENERALIZED INEQUALITY 191
To the i th extreme direction v = Γ i ∈ R n of cone K , ascribe a coordinate
t ⋆ i(x)∈ R of x from definition (432). On domain K , the mapping
⎡ ⎤
t ⋆ 1(x)
t ⋆ ⎢ ⎥
(x) = ⎣ . ⎦: R n → R N (433)
t ⋆ N (x)
has no nontrivial nullspace. Because x −t ⋆ v must belong to ∂K by definition,
the mapping t ⋆ (x) is equivalent to a convex problem (separable in index i)
whose objective (by (298)) is tightly bounded below by 0 :
t ⋆ (x) ≡ arg minimize
t∈R N
N∑
i=1
Γ ∗T
j(i) (x − t iΓ i )
subject to x − t i Γ i ∈ K ,
i=1... N
(434)
where index j ∈ I is dependent on i and where (by (331)) λ = Γj ∗ ∈ R n is an
extreme direction of dual cone K ∗ that is normal to a hyperplane supporting
K and containing x − t ⋆ iΓ i . Because extreme-direction cardinality N for
cone K is not necessarily the same as for dual cone K ∗ , index j must be
judiciously selected from a set I .
To prove injectivity when extreme-direction cardinality N > n exceeds
spatial dimension, we need only show mapping t ⋆ (x) to be invertible;
[106, thm.9.2.3] id est, x is recoverable given t ⋆ (x) :
N∑
x = arg minimize Γ
x∈R n j(i) ∗T(x
− t⋆ iΓ i )
i=1
(435)
subject to x − t ⋆ iΓ i ∈ K , i=1... N
The feasible set of this nonseparable convex problem is an intersection of
translated full-dimensional pointed closed convex cones ⋂ i K + t⋆ iΓ i . The
objective function’s linear part describes movement in normal-direction −Γj
∗
for each of N hyperplanes. The optimal point of hyperplane intersection is
the unique solution x when {Γj ∗ } comprises n linearly independent normals
that come from the dual cone and make the objective vanish. Because the
dual cone K ∗ is full-dimensional, pointed, closed, and convex by assumption,
there exist N extreme directions {Γj ∗ } from K ∗ ⊂ R n that span R n . So
we need simply choose N spanning dual extreme directions that make the
optimal objective vanish. Because such dual extreme directions preexist
by (298), t ⋆ (x) is invertible.
Otherwise, in the case N ≤ n , t ⋆ (x) holds coordinates of biorthogonal
expansion. Reconstruction of x is therefore unique.