v2009.01.01 - Convex Optimization

E.5. PROJECTION EXAMPLES 657
a ∗ 2
K ∗
a 2
a ∗ 1
z
x
K
0
y
a 1
a 1 ⊥ a ∗ 2
a 2 ⊥ a ∗ 1
x = y + z = P a1 x + P a2 x
K ∗
Figure 139: (confer Figure 55) Biorthogonal expansion of point x∈aff K is
found by projecting x nonorthogonally on extreme directions of polyhedral
cone K ⊂ R 2 . (Dotted lines of projection bound this translated negated cone.)
Direction of projection on extreme direction a 1 is orthogonal to extreme
direction a ∗ 1 of dual cone K ∗ and parallel to a 2 (E.3.5); similarly, direction
of projection on a 2 is orthogonal to a ∗ 2 and parallel to a 1 . Point x is
sum of nonorthogonal projections: x on R(a 1 ) and x on R(a 2 ). Expansion
is unique because extreme directions of K are linearly independent. Were
a 1 orthogonal to a 2 , then K would be identical to K ∗ and nonorthogonal
projections would become orthogonal.