v2009.01.01 - Convex Optimization

6.8. DUAL EDM CONE 487
has dual affine dimension complementary to affine dimension corresponding
to the optimal solution of
minimize ‖D − H‖ F
D∈S N h
subject to −VN TDV N ≽ 0
(1189)
Precisely,
rank(D ◦⋆ −δ(D ◦⋆ 1)) + rank(V T ND ⋆ V N ) = N −1 (1190)
and rank(D ◦⋆ −δ(D ◦⋆ 1))≤N−1 because vector 1 is always in the nullspace
of rank’s argument. This is similar to the known result for projection on the
self-dual positive semidefinite cone and its polar:
rankP −S N
+
H + rankP S N
+
H = N (1191)
When low affine dimension is a desirable result of projection on the
EDM cone, projection on the polar EDM cone should be performed
instead. Convex polar problem (1188) can be solved for D ◦⋆ by
transforming to an equivalent Schur-form semidefinite program (3.1.7.2).
Interior-point methods for numerically solving semidefinite programs tend
to produce high-rank solutions. (4.1.1.1) Then D ⋆ = H − D ◦⋆ ∈ EDM N by
Corollary E.9.2.2.1, and D ⋆ will tend to have low affine dimension. This
approach breaks when attempting projection on a cone subset discriminated
by affine dimension or rank, because then we have no complementarity
relation like (1190) or (1191) (7.1.4.1).
Lemma 6.8.1.1.1; rewriting,
minimize ‖(D ◦ − δ(D ◦ 1)) − (H − δ(D ◦ 1))‖ F
D ◦ ∈ S N
subject to D ◦ − δ(D ◦ 1) ≽ 0
which is the projection of affinely transformed optimal solution H − δ(D ◦⋆ 1) on S N c ∩ S N + ;
D ◦⋆ − δ(D ◦⋆ 1) = P S N
+
P S N
c
(H − δ(D ◦⋆ 1))
Foreknowledge of an optimal solution D ◦⋆ as argument to projection suggests recursion.