v2009.01.01 - Convex Optimization

684 APPENDIX E. PROJECTION
Projection of x on the set of all cardinality-k vectors {y | card y ≤k}
keeps k entries of greatest magnitude and clips to 0 those remaining.
Unique minimum-distance projection of H ∈ S n on the positive
semidefinite cone S n + in the Euclidean/Frobenius sense is accomplished
by eigen decomposition (diagonalization) followed by clipping all
negative eigenvalues to 0.
Unique minimum-distance projection on the generally nonconvex
subset of all matrices belonging to S n + having rank not exceeding ρ
(2.9.2.1) is accomplished by clipping all negative eigenvalues to 0 and
zeroing the smallest nonnegative eigenvalues keeping only ρ largest.
(7.1.2)
Unique minimum-distance projection, in the Euclidean/Frobenius
sense, of H ∈ R m×n on the generally nonconvex subset of all m ×n
matrices of rank no greater than k is the singular value decomposition
(A.6) of H having all singular values beyond the k th zeroed.
This is also a solution to projection in the sense of spectral norm.
[285, p.79, p.208]
Projection on K of any point x∈−K ∗ , belonging to the polar cone, is
equivalent to projection on the origin. (E.9.2)
P S N
+ ∩ S N c
= P S N
+
P S N c
(1169)
P R
N×N
+ ∩ S N h
= P R
N×NP + S N
h
P R
N×N
+ ∩ S
= P N R
N×N
+
P S N
(7.0.1.1)
(E.9.5)
E.9.4.0.1 Exercise. Largest singular value.
Find the unique minimum-distance projection on the set of all m ×n
matrices whose largest singular value does not exceed 1.
E.9.4.1
notes
Projection on Lorentz (second-order) cone: [53, exer.8.3(c)].
Deutsch [95] provides an algorithm for projection on polyhedral cones.
Youla [343,2.5] lists eleven “useful projections”, of square-integrable
uni- and bivariate real functions on various convex sets, in closed form.
Unique minimum-distance projection on an ellipsoid: Example 4.6.0.0.1.