v2007.11.26 - Convex Optimization

7.2. SECOND PREVALENT PROBLEM: 475So convex problem (1187) is equivalent to the semidefinite programminimize − tr(V (D − 2Y )V )D , Y[ ]dij y ijsubject to≽ 0 ,j > i = 1... N −1y ij h 2 ij(1191)Y ∈ S N hD ∈ EDM Nwhere the constants h 2 ij and N have been dropped arbitrarily from theobjective.7.2.1.2 Gram-form semidefinite program, Problem 2, convex caseThere is great advantage to expressing problem statement (1191) inGram-form because Gram matrix G is a bidirectional bridge between pointlist X and distance matrix D ; e.g., Example 5.4.2.2.4, Example 6.4.0.0.1.This way, problem convexity can be maintained while simultaneouslyconstraining point list X , Gram matrix G , and distance matrix D at ourdiscretion.Convex problem (1191) may be equivalently written via linear bijective(5.6.1) EDM operator D(G) (746);minimizeG∈S N c , Y ∈ SN hsubject to− tr(V (D(G) − 2Y )V )[ ]〈Φij , G〉 y ij≽ 0 ,y ijh 2 ijj > i = 1... N −1(1192)G ≽ 0where distance-square D = [d ij ] ∈ S N h (730) is related to Gram matrix entriesG = [g ij ] ∈ S N c ∩ S N + byd ij = g ii + g jj − 2g ij= 〈Φ ij , G〉(745)whereΦ ij = (e i − e j )(e i − e j ) T ∈ S N + (732)