v2009.01.01 - Convex Optimization

2.13. DUAL CONE & GENERALIZED INEQUALITY 167
an orthogonal set; neither do extreme directions Γ 3 and Γ 4 of dual cone K ∗ ;
rather, we have the biorthogonality condition, [318]
Γ T 4 Γ 1 = Γ T 3 Γ 2 = 0
Γ T 3 Γ 1 ≠ 0, Γ T 4 Γ 2 ≠ 0
(348)
Biorthogonal expansion of x ∈ K is then
x = Γ 1
Γ T 3 x
Γ T 3 Γ 1
+ Γ 2
Γ T 4 x
Γ T 4 Γ 2
(349)
where Γ T 3 x/(Γ T 3 Γ 1 ) is the nonnegative coefficient of nonorthogonal projection
(E.6.1) of x on Γ 1 in the direction orthogonal to Γ 3 (y in Figure 139, p.657),
and where Γ T 4 x/(Γ T 4 Γ 2 ) is the nonnegative coefficient of nonorthogonal
projection of x on Γ 2 in the direction orthogonal to Γ 4 (z in Figure 139);
they are coordinates in this nonorthogonal system. Those coefficients must
be nonnegative x ≽ K
0 because x ∈ K (293) and K is simplicial.
If we ascribe the extreme directions of K to the columns of a matrix
X ∆ = [ Γ 1 Γ 2 ] (350)
then we find that the pseudoinverse transpose matrix
[
]
X †T 1 1
= Γ 3 Γ
Γ T 4
3 Γ 1 Γ T 4 Γ 2
(351)
holds the extreme directions of the dual cone. Therefore,
x = XX † x (357)
is the biorthogonal expansion (349) (E.0.1), and the biorthogonality
condition (348) can be expressed succinctly (E.1.1) 2.61
X † X = I (358)
Expansion w=XX † w for any w ∈ R 2 is unique if and only if the extreme
directions of K are linearly independent; id est, iff X has no nullspace.
2.61 Possibly confusing is the fact that formula XX † x is simultaneously the orthogonal
projection of x on R(X) (1786), and a sum of nonorthogonal projections of x ∈ R(X) on
the range of each and every column of full-rank X skinny-or-square (E.5.0.0.2).