v2006.03.09 - Convex Optimization

7.3. THIRD PREVALENT PROBLEM: 4137.3 Third prevalent problem:Projection on EDM cone in d ijReformulating Problem 2 (p.405) in terms of EDM D changes the problemconsiderably:⎫minimize ‖D − H‖ 2 F ⎪⎬Dsubject to rankVN TDV N ≤ ρ Problem 3 (999)⎪D ∈ EDM N ⎭This third prevalent proximity problem is a Euclidean projection of givenmatrix H on a generally nonconvex subset (when ρ < N −1) of theboundary of the convex cone of Euclidean distance matrices ∂EDM N relativeto subspace S N h (Figure 74(d)). Because coordinates of projection aredistance-square and H presumably now holds distance-square measurements,numerical solution to Problem 3 is generally different than that of Problem 2.For the moment, we need make no assumptions regarding measurementmatrix H .7.3.1 Convex caseminimize ‖D − H‖ 2 FD(1000)subject to D ∈ EDM NWhen the rank constraint disappears (for ρ = N −1), this third problembecomes obviously convex because the feasible set is then the entire EDMcone and because the objective function‖D − H‖ 2 F = ∑ i,j(d ij − h ij ) 2 (1001)is a strictly convex quadratic in D ; 7.167.16 For nonzero Y ∈ S N h and some open interval of t∈R (3.1.2.3.2,D.2.3)d 2dt 2 ‖(D + tY ) − H‖2 F = 2 tr Y T Y > 0