v2009.01.01 - Convex Optimization

E.9. PROJECTION ON CONVEX SET 683
The foregoing proof reveals another flavor of nonexpansivity; for each and
every x,y ∈ R n
‖Px − Py‖ 2 + ‖(I − P )x − (I − P )y‖ 2 ≤ ‖x − y‖ 2 (1916)
Deutsch shows yet another: [92,5.5]
E.9.4
‖Px − Py‖ 2 ≤ 〈x − y , Px − Py〉 (1917)
Easy projections
Projecting any matrix H ∈ R n×n in the Euclidean/Frobenius sense
orthogonally on the subspace of symmetric matrices S n in isomorphic
R n2 amounts to taking the symmetric part of H ; (2.2.2.0.1) id est,
(H+H T )/2 is the projection.
To project any H ∈ R n×n orthogonally on the symmetric hollow
subspace S n h in isomorphic Rn2 (2.2.3.0.1), we may take the symmetric
part and then zero all entries along the main diagonal, or vice versa
(because this is projection on the intersection of two subspaces); id est,
(H + H T )/2 − δ 2 (H) .
To project a matrix on the nonnegative orthant R m×n
+ , simply clip all
negative entries to 0. Likewise, projection on the nonpositive orthant
R m×n
−
sees all positive entries clipped to 0. Projection on other orthants
is equally simple with appropriate clipping.
Projecting on hyperplane, halfspace, slab:E.5.0.0.8.
Projection of y ∈ R n on Euclidean ball B = {x∈ R n | ‖x − a‖ ≤ c} :
for y ≠ a , P B y = (y − a) + a .
c
‖y−a‖
Clipping in excess of |1| each entry of a point x∈ R n is equivalent to
unique minimum-distance projection of x on a hypercube centered at
the origin. (conferE.10.3.2)
Projection of x∈ R n on a (rectangular) hyperbox: [53,8.1.1]
C = {y ∈ R n | l ≼ y ≼ u, l ≺ u} (1918)
⎧
⎨ l k , x k ≤ l k
P(x) k=0...n = x k , l k ≤ x k ≤ u k (1919)
⎩
u k , x k ≥ u k