v2007.11.26 - Convex Optimization

7.3. THIRD PREVALENT PROBLEM: 483found by searching 6! local minima of (1208) [70]. By iterating convexproblems (1210) and (1211) about twenty times (initial W = 0) we find theglobal infimum 98.12812 of stress problem (1208), and by (965) we find acorresponding one-dimensional point list that is a rigid transformation in Rof X ⋆ .Here we found the infimum to accuracy of the given data, but that ceasesto hold as problem size increases. Because of machine numerical precisionand an interior-point method of solution, we speculate, accuracy degradesquickly as problem size increases beyond this.7.3 Third prevalent problem:Projection on EDM cone in d ijReformulating Problem 2 (p.472) in terms of EDM D changes the problemconsiderably:⎫minimize ‖D − H‖ 2 F ⎪⎬Dsubject to rankVN TDV N ≤ ρ Problem 3 (1214)⎪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 relative tosubspace S N h (Figure 100(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(1215)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 EDM