v2009.01.01 - Convex Optimization

512 CHAPTER 7. PROXIMITY PROBLEMS
R N−1
+ requires: (E.9.2.0.1)
δ(Υ ⋆ ) ρ+1:N−1 = 0
δ(Υ ⋆ ) ≽ 0
δ(Υ ⋆ ) T( δ(Υ ⋆ ) − π(δ(R T ΛR)) ) = 0
δ(Υ ⋆ ) − π(δ(R T ΛR)) ≽ 0
(1245)
which are necessary and sufficient conditions. Any value Υ ⋆ satisfying
conditions (1245) is optimal for (1244a). So
{ {
δ(Υ ⋆ max 0, π ( δ(R
) i =
T ΛR) ) }
, i=1... ρ
i
(1246)
0, i=ρ+1... N −1
specifies an optimal solution. The lower bound on the objective with respect
to R in (1244b) is tight; by (1212)
‖ |Υ ⋆ | − |Λ| ‖ F ≤ ‖Υ ⋆ − R T ΛR‖ F (1247)
where | | denotes absolute entry-value. For selection of Υ ⋆ as in (1246), this
lower bound is attained when (conferC.4.2.2)
R ⋆ = I (1248)
which is the known solution.
7.1.4.1 Significance
Importance of this well-known [107] optimal solution (1221) for projection
on a rank ρ subset of a positive semidefinite cone should not be dismissed:
Problem 1, as stated, is generally nonconvex. This analytical solution
at once encompasses projection on a rank ρ subset (237) of the positive
semidefinite cone (generally, a nonconvex subset of its boundary)
from either the exterior or interior of that cone. 7.9 By problem
transformation to the spectral domain, projection on a rank ρ subset
becomes a convex optimization problem.
7.9 Projection on the boundary from the interior of a convex Euclidean body is generally
a nonconvex problem.