v2010.10.26 - Convex Optimization

484 CHAPTER 5. EUCLIDEAN DISTANCE MATRIX5.13.2.2 Isotonic solution with sort constraintBecause problems involving rank are generally difficult, we will partition(1147) 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)(1149)where sort-index matrix O (a given constant in (a)) becomes an implicitvector variable o i solving the i th instance of (1149b)1√2dvec O i = o i Π T σ ⋆ ∈ R N(N−1)/2 , i∈{1, 2, 3...} (1150)As mentioned in discussion of relaxed problem (1148), a closed-formsolution to problem (1149a) exists. Only the first iteration of (1149a)sees the original sort-index matrix O whose entries are nonnegative wholenumbers; id est, O 0 =O∈ S N h ∩ R N×N+ (1144). Subsequent iterations i takethe previous solution of (1149b) as inputO i = dvec −1 ( √ 2o i ) ∈ S N (1151)real successors, estimating distance-square not order, to the sort-indexmatrix O .New convex problem (1149b) finds the unique minimum-distanceprojection of Πd 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 (430)where mN(N −1)/2, we may rewrite (1149b) as an equivalent quadraticprogram; a convex problem in terms of the halfspace-description of K M+ :minimize (σ − Πd) T (σ − Πd)σsubject to Y † σ ≽ 0(1152)