v2009.01.01 - Convex Optimization

702 APPENDIX E. PROJECTION
K ⊥ H 1 ∩ H 2
(0)
K ⊥ H 1 ∩ H 2
(Pb) + Pb
H 1
0
H 2
K ∆ = H 1 ∩ H 2
Pb
b
Figure 149: Two examples (truncated): Normal cone to H 1 ∩ H 2 at the
origin, and at point Pb on the boundary. H 1 and H 2 are the same halfspaces
from Figure 148. The normal cone at the origin K ⊥ H 1 ∩ H 2
(0) is simply −K ∗ .
E.10.3.2.1 Definition. Normal cone. [231] [37, p.261] [173,A.5.2]
[48,2.1] [265,3] The normal cone to any set S ⊆ R n at any particular
point a∈ R n is defined as the closed cone
K ⊥ S (a) ∆ = {z ∈ R n | z T (y −a)≤0 ∀y ∈ S} = −(S − a) ∗ (1964)
an intersection of halfspaces about the origin in R n hence convex regardless
of the convexity of S ; the negative dual cone to the translate S − a . △
Examples of normal cone construction are illustrated in Figure 149: The
normal cone at the origin is the vector sum (2.1.8) of two normal cones;
[48,3.3, exer.10] for H 1 ∩ int H 2 ≠ ∅
K ⊥ H 1 ∩ H 2
(0) = K ⊥ H 1
(0) + K ⊥ H 2
(0) (1965)
This formula applies more generally to other points in the intersection.
The normal cone to any affine set A at α∈ A , for example, is the
orthogonal complement of A − α . Projection of any point in the translated
normal cone KC ⊥ (a∈ C) + a on convex set C is identical to a ; in other words,
point a is that point in C closest to any point belonging to the translated
normal cone KC ⊥ (a) + a ; e.g., Theorem E.4.0.0.1.