v2009.01.01 - Convex Optimization

E.9. PROJECTION ON CONVEX SET 681
E.9.2.2.2 Corollary. Unique projection via dual or normal cone.
[92,4.7] (E.10.3.2, confer Theorem E.9.1.0.3) Given point x∈ R n and
closed convex cone K ⊆ R n , the following are equivalent statements:
1. point Px is the unique minimum-distance projection of x on K
2. Px ∈ K , x − Px ∈ −(K − Px) ∗ = −K ∗ ∩ (Px) ⊥
3. Px ∈ K , 〈x − Px, Px〉 = 0, 〈x − Px, y〉 ≤ 0 ∀y ∈ K
⋄
E.9.2.2.3 Example. Unique projection on nonnegative orthant.
(confer (1225)) From Theorem E.9.2.0.1, to project matrix H ∈ R m×n on
the self-dual orthant (2.13.5.1) of nonnegative matrices R m×n
+ in isomorphic
R mn , the necessary and sufficient conditions are:
H ⋆ ≥ 0
tr ( (H ⋆ − H) T H ⋆) = 0
H ⋆ − H ≥ 0
(1908)
where the inequalities denote entrywise comparison. The optimal solution
H ⋆ is simply H having all its negative entries zeroed;
H ⋆ ij = max{H ij , 0} , i,j∈{1... m} × {1... n} (1909)
Now suppose the nonnegative orthant is translated by T ∈ R m×n ; id est,
consider R m×n
+ + T . Then projection on the translated orthant is [92,4.8]
H ⋆ ij = max{H ij , T ij } (1910)
E.9.2.2.4 Example. Unique projection on truncated convex cone.
Consider the problem of projecting a point x on a closed convex cone that
is artificially bounded; really, a bounded convex polyhedron having a vertex
at the origin:
minimize ‖x − Ay‖ 2
y∈R N
subject to y ≽ 0
(1911)
‖y‖ ∞ ≤ 1