v2010.10.26 - Convex Optimization

7.2. SECOND PREVALENT PROBLEM: 5757.2.2.5 Applying log det rank-heuristic to Problem 2When the log det rank-heuristic is inserted into Problem 2, problem (1370)becomes the problem sequence in iminimizeD , Y , κsubject toκρ + 2 tr(V Y V )[ ]djl y jl≽ 0 ,y jlh 2 jll > j = 1... N −1− tr((−V D i V + εI) −1 V DV ) ≤ κρ(1377)Y ∈ S N hD ∈ EDM Nwhere D i+1 D ⋆ ∈ EDM N and D 0 11 T − I .7.2.2.6 Tightening this log det rank-heuristicLike the trace method, this log det technique for constraining rank offersno provision for meeting a predetermined upper bound ρ . Yet sinceeigenvalues are simply determined, λ(Y i + εI)=δ(Λ + εI) , we may certainlyforce selected weights to ε −1 by manipulating diagonalization (1375a).Empirically we find this sometimes leads to better results, although affinedimension of a solution cannot be guaranteed.7.2.2.7 Cumulative summary of rank heuristicsWe have studied a perturbation method of rank reduction in4.3 as well asthe trace heuristic (convex envelope method7.2.2.1.1) and log det heuristicin7.2.2.4. There is another good contemporary method called LMIRank[285] based on alternating projection (E.10). 7.177.2.2.7.1 Example. Unidimensional scaling.We apply the convex iteration method from4.4.1 to numerically solve aninstance of Problem 2; a method empirically superior to the foregoing convexenvelope and log det heuristics for rank regularization and enforcing affinedimension.7.17 that does not solve the ball packing problem presented in5.4.2.3.4.