v2010.10.26 - Convex Optimization

549project on the subspace, then project the result on the orthant in thatsubspace. (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, 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(1302)a convex optimization problem. Because the symmetric hollow subspace S N his orthogonal to the antisymmetric antihollow subspace R N×N⊥h(2.2.3), thenfor B ∈ S N h( ( ))1tr B T 2 (H −HT ) + δ 2 (H) = 0 (1303)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(1304)This means the antisymmetric antihollow part of given matrix H wouldbe ignored by minimization with respect to symmetric hollow variable Bunder Frobenius’ norm; id est, minimization proceeds as though given thesymmetric hollow part of H .This action of Frobenius’ norm (1304) is effectively a Euclidean projection(minimum-distance projection) of H on the symmetric hollow subspace S N hprior to minimization. Thus minimization proceeds inherently following thecorrect order for projection on S N h ∩ C . Therefore we may either assumeH ∈ 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.