v2010.10.26 - Convex Optimization

576 CHAPTER 7. PROXIMITY PROBLEMSUnidimensional scaling, [105] a historically practical application ofmultidimensional scaling (5.12), entails solution of an optimization problemhaving local minima whose multiplicity varies as the factorial of point-listcardinality; geometrically, it means reconstructing a list constrained to lie inone affine dimension. In terms of point list, the nonconvex problem is: givennonnegative symmetric matrix H = [h ij ] ∈ S N ∩ R N×N+ (1357) whose entriesh ij are all known,minimize{x i ∈R}N∑(|x i − x j | − h ij ) 2 (1308)i , j=1called a raw stress problem [53, p.34] which has an implicit constraint ondimensional embedding of points {x i ∈ R , i=1... N}. This problem hasproven NP-hard; e.g., [73].As always, we first transform variables to distance-square D ∈ S N h ; sobegin with convex problem (1359) on page 569minimize − tr(V (D − 2Y )V )D , Y[ ]dij y ijsubject to≽ 0 ,y ijh 2 ijY ∈ S N hD ∈ EDM NrankV T N DV N = 1N ≥ j > i = 1... N −1(1378)that becomes equivalent to (1308) by making explicit the constraint on affinedimension via rank. The iteration is formed by moving the dimensionalconstraint to the objective:minimize −〈V (D − 2Y )V , I〉 − w〈VN TDV N , W 〉D , Y[ ]dij y ijsubject to≽ 0 , N ≥ j > i = 1... N −1y ij h 2 ij(1379)Y ∈ S N hD ∈ EDM Nwhere w (≈ 10) is a positive scalar just large enough to make 〈V T N DV N , W 〉vanish to within some numerical precision, and where direction matrix W is