v2010.10.26 - Convex Optimization

578 CHAPTER 7. PROXIMITY PROBLEMS7.3 Third prevalent problem:Projection on EDM cone in d ijIn summary, we find that the solution to problem [(1310.3) p.553]is difficult and depends on the dimension of the space as thegeometry of the cone of EDMs becomes more complex.−Hayden, Wells, Liu, & Tarazaga (1991) [185,3]Reformulating Problem 2 (p.566) in terms of EDM D changes theproblem considerably:⎫minimize ‖D − H‖ 2 F ⎪⎬Dsubject to rankVN TDV N ≤ ρ Problem 3 (1383)⎪D ∈ EDM N ⎭This third prevalent proximity problem is a Euclidean projection of givenmatrix H on a generally nonconvex subset (ρ < N −1) of ∂EDM N theboundary of the convex cone of Euclidean distance matrices relativeto subspace S N h (Figure 138d). Because coordinates of projection aredistance-square and H now presumably 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(1384)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 (1385)