v2006.03.09 - Convex Optimization

408 CHAPTER 7. EDM PROXIMITY7.2.1.2 Gram-form semidefinite program, Problem 2, convex caseThere is great advantage to expressing problem statement (985) inGram-form because Gram matrix G is a bidirectional bridge between pointlist X and distance matrix D ; e.g., Example 4.4.2.2.4, Example 5.4.0.0.1.This way, problem convexity can be maintained while simultaneouslyconstraining point list X , Gram matrix G , and distance matrix D at ourdiscretion.Convex problem (985) may be equivalently written via linear bijective(4.6.1) EDM operator D(G) (472);minimizeG∈S N c , Y ∈ S N hsubject to− tr(V (D(G) − 2Y )V )[ ]〈Φij , G〉 y ij≽ 0 ,y ijh 2 ijj > i = 1... N −1(986)G ≽ 0where distance-square D = [d ij ] ∈ S N h (456) is related to Gram matrix entriesG = [g ij ] ∈ S N c ∩ S N + byd ij = g ii + g jj − 2g ij= 〈Φ ij , G〉(471)whereΦ ij = (e i − e j )(e i − e j ) T ∈ S N + (458)Confinement of G to the geometric center subspace provides numericalstability and no loss of generality (confer (733)); implicit constraint G1 = 0is otherwise unnecessary.To include constraints on the list X ∈ R n×N , we would first rewrite (986)minimize − tr(V (D(G) − 2Y )V )G∈S N c , Y ∈ SN h , X∈ Rn×N [ ]〈Φij , G〉 y ijsubject to≽ 0 ,X Ty ijh 2 ij[ ] I X≽ 0GX ∈ Cj > i = 1... N −1(987)