v2009.01.01 - Convex Optimization

7.3. THIRD PREVALENT PROBLEM: 525
global infimum 98.12812 of stress problem (1213), and by (1037) we find a
corresponding one-dimensional point list that is a rigid transformation in R
of X ⋆ .
Here we found the infimum to accuracy of the given data, but that ceases
to hold as problem size increases. Because of machine numerical precision
and an interior-point method of solution, we speculate, accuracy degrades
quickly as problem size increases beyond this.
7.3 Third prevalent problem:
Projection on EDM cone in d ij
Reformulating Problem 2 (p.514) in terms of EDM D changes the problem
considerably:
⎫
minimize ‖D − H‖ 2 F ⎪⎬
D
subject to rankVN TDV N ≤ ρ Problem 3 (1286)
⎪
D ∈ EDM N ⎭
This third prevalent proximity problem is a Euclidean projection of given
matrix H on a generally nonconvex subset (ρ < N −1) of ∂EDM N the
boundary of the convex cone of Euclidean distance matrices relative to
subspace S N h (Figure 114(d)). Because coordinates of projection are
distance-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 measurement
matrix H .
7.3.1 Convex case
minimize ‖D − H‖ 2 F
D
(1287)
subject to D ∈ EDM N
When the rank constraint disappears (for ρ = N −1), this third problem
becomes obviously convex because the feasible set is then the entire EDM
cone and because the objective function
‖D − H‖ 2 F = ∑ i,j
(d ij − h ij ) 2 (1288)