v2009.01.01 - Convex Optimization

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project on the subspace, then project the result on the orthant in that
subspace. (conferE.9.5)
In contrast, order of projection on an intersection of subspaces is arbitrary.
That order-of-projection rule applies more generally, of course, to
the intersection of any convex set C with any subspace. Consider the
proximity problem 7.1 over convex feasible set S N h ∩ C given nonsymmetric
nonhollow H ∈ R N×N :
minimize
B∈S N h
subject to
‖B − H‖ 2 F
B ∈ C
(1207)
a convex optimization problem. Because the symmetric hollow subspace
is orthogonal to the antisymmetric antihollow subspace (2.2.3), then for
B ∈ S N h
( ( ))
1
tr B T 2 (H −HT ) + δ 2 (H) = 0 (1208)
so the objective function is equivalent to
( )∥ ‖B − H‖ 2 F ≡
1 ∥∥∥
2
∥ B − 2 (H +HT ) − δ 2 (H) +
F
2
1
∥2 (H −HT ) + δ 2 (H)
∥
F
(1209)
This means the antisymmetric antihollow part of given matrix H would
be ignored by minimization with respect to symmetric hollow variable B
under the Frobenius norm; id est, minimization proceeds as though given
the symmetric hollow part of H .
This action of the Frobenius norm (1209) is effectively a Euclidean
projection (minimum-distance projection) of H on the symmetric hollow
subspace S N h prior to minimization. Thus minimization proceeds inherently
following the correct order for projection on S N h ∩ C . Therefore we
may either assume H ∈ S N h , or take its symmetric hollow part prior to
optimization.
7.1 There are two equivalent interpretations of projection (E.9): one finds a set normal,
the other, minimum distance between a point and a set. Here we realize the latter view.