v2007.11.26 - Convex Optimization

390 CHAPTER 5. EUCLIDEAN DISTANCE MATRIXThat sort constraint demands: any optimal solution D ⋆ must possess theknown comparative distance relationship that produces the original ordinaldistance data O (977). Ignoring the sort constraint, apparently, violates it.Yet even more remarkable is how much the map reconstructed using onlyordinal data still resembles the original map of the USA after suffering themany violations produced by solving relaxed problem (981). This suggeststhe simple reconstruction techniques of5.12 are robust to a significantamount of noise.5.13.2.2 Isotonic solution with sort constraintBecause problems involving rank are generally difficult, we will partition(980) into two problems we know how to solve and then alternate theirsolution until convergence:minimize ‖−VN T(D − O)V N ‖ FDsubject to rankVN TDV N ≤ 3D ∈ EDM Nminimize ‖σ − Πd‖σsubject to σ ∈ K M+(a)(b)(982)where the sort-index matrix O (a given constant in (a)) becomes an implicitvectorized variable o i solving the i th instance of (982b)o i ∆ = Π T σ ⋆ = 1 √2dvecO i ∈ R N(N−1)/2 , i∈{1, 2, 3...} (983)As mentioned in discussion of relaxed problem (981), a closed-formsolution to problem (982a) exists. Only the first iteration of (982a) sees theoriginal sort-index matrix O whose entries are nonnegative whole numbers;id est, O 0 =O∈ S N h ∩ R N×N+ (977). Subsequent iterations i take the previoussolution of (982b) as inputO i = dvec −1 ( √ 2o i ) ∈ S N (984)real successors to the sort-index matrix O .New problem (982b) finds the unique minimum-distance projection ofΠd on the monotone nonnegative cone K M+ . By definingY †T ∆ = [e 1 − e 2 e 2 −e 3 e 3 −e 4 · · · e m ] ∈ R m×m (372)