v2007.09.13 - Convex Optimization

In contrast, order of projection on an intersection of subspaces is arbitrary.439That order-of-projection rule applies more generally, of course, tothe intersection of any convex set C with any subspace. Consider theproximity problem 7.1 over convex feasible set S N h ∩ C given nonsymmetricnonhollow H ∈ R N×N :minimizeB∈S N hsubject to‖B − H‖ 2 FB ∈ C(1104)a convex optimization problem. Because the symmetric hollow subspaceis orthogonal to the antisymmetric antihollow subspace (2.2.3), then forB ∈ S N h( ( ))1tr B T 2 (H −HT ) + δ 2 (H) = 0 (1105)so the objective function is equivalent to( )∥ ‖B − H‖ 2 F ≡1 ∥∥∥2∥ B − 2 (H +HT ) − δ 2 (H) +F21∥2 (H −HT ) + δ 2 (H)∥F(1106)This means the antisymmetric antihollow part of given matrix H wouldbe ignored by minimization with respect to symmetric hollow variable Bunder the Frobenius norm; id est, minimization proceeds as though giventhe symmetric hollow part of H .This action of the Frobenius norm (1106) is effectively a Euclideanprojection (minimum-distance projection) of H on the symmetric hollowsubspace S N h prior to minimization. Thus minimization proceeds inherentlyfollowing the correct order for projection on S N h ∩ C . Therefore wemay either assume H ∈ S N h , or take its symmetric hollow part prior tooptimization.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.