v2009.01.01 - Convex Optimization

7.2. SECOND PREVALENT PROBLEM: 517
So convex problem (1260) is equivalent to the semidefinite program
minimize − tr(V (D − 2Y )V )
D , Y
[ ]
dij y ij
subject to
≽ 0 ,
N ≥ j > i = 1... N −1
y ij h 2 ij
(1264)
Y ∈ S N h
D ∈ EDM N
where the constants h 2 ij and N have been dropped arbitrarily from the
objective.
7.2.1.2 Gram-form semidefinite program, Problem 2, convex case
There is great advantage to expressing problem statement (1264) in
Gram-form because Gram matrix G is a bidirectional bridge between point
list 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 simultaneously
constraining point list X , Gram matrix G , and distance matrix D at our
discretion.
Convex problem (1264) may be equivalently written via linear bijective
(5.6.1) EDM operator D(G) (810);
minimize
G∈S N c , Y ∈ SN h
subject to
− tr(V (D(G) − 2Y )V )
[ ]
〈Φij , G〉 y ij
≽ 0 ,
y ij
h 2 ij
N ≥ j > i = 1... N −1
(1265)
G ≽ 0
where distance-square D = [d ij ] ∈ S N h (794) is related to Gram matrix entries
G = [g ij ] ∈ S N c ∩ S N + by
d ij = g ii + g jj − 2g ij
= 〈Φ ij , G〉
(809)
where
Φ ij = (e i − e j )(e i − e j ) T ∈ S N + (796)