v2009.01.01 - Convex Optimization

170 CHAPTER 2. CONVEX GEOMETRY
2.13.8 Biorthogonal expansion, derivation
Biorthogonal expansion is a means for determining coordinates in a pointed
conic coordinate system characterized by a nonorthogonal basis. Study
of nonorthogonal bases invokes pointed polyhedral cones and their duals;
extreme directions of a cone K are assumed to constitute the basis while
those of the dual cone K ∗ determine coordinates.
Unique biorthogonal expansion with respect to K depends upon existence
of its linearly independent extreme directions: Polyhedral cone K must be
pointed; then it possesses extreme directions. Those extreme directions must
be linearly independent to uniquely represent any point in their span.
We consider nonempty pointed polyhedral cone K having possibly empty
interior; id est, we consider a basis spanning a subspace. Then we need
only observe that section of dual cone K ∗ in the affine hull of K because, by
expansion of x , membership x∈aff K is implicit and because any breach
of the ordinary dual cone into ambient space becomes irrelevant (2.13.9.3).
Biorthogonal expansion
x = XX † x ∈ aff K = aff cone(X) (357)
is expressed in the extreme directions {Γ i } of K arranged columnar in
under assumption of biorthogonality
X = [ Γ 1 Γ 2 · · · Γ N ] ∈ R n×N (253)
X † X = I (358)
where † denotes matrix pseudoinverse (E). We therefore seek, in this
section, a vertex-description for K ∗ ∩ aff K in terms of linearly independent
dual generators {Γ ∗ i }⊂aff K in the same finite quantity 2.63 as the extreme
directions {Γ i } of
K = cone(X) = {Xa | a ≽ 0} ⊆ R n (94)
We assume the quantity of extreme directions N does not exceed the
dimension n of ambient vector space because, otherwise, the expansion could
not be unique; id est, assume N linearly independent extreme directions
2.63 When K is contained in a proper subspace of R n , the ordinary dual cone K ∗ will have
more generators in any minimal set than K has extreme directions.