v2009.01.01 - Convex Optimization

526 CHAPTER 7. PROXIMITY PROBLEMS
is a strictly convex quadratic in D ; 7.16
∑
minimize d 2 ij − 2h ij d ij + h 2 ij
D
i,j
(1289)
subject to D ∈ EDM N
Optimal solution D ⋆ is therefore unique, as expected, for this simple
projection on the EDM cone equivalent to (1214).
7.3.1.1 Equivalent semidefinite program, Problem 3, convex case
In the past, this convex problem was solved numerically by means of
alternating projection. (Example 7.3.1.1.1) [130] [123] [159,1] We translate
(1289) to an equivalent semidefinite program because we have a good solver:
Assume the given measurement matrix H to be nonnegative and
symmetric; 7.17
H = [h ij ] ∈ S N ∩ R N×N
+ (1262)
We then propose: Problem (1289) is equivalent to the semidefinite program,
for
∂ = ∆ [d 2 ij ] = D ◦D (1290)
a matrix of distance-square squared,
minimize − tr(V (∂ − 2H ◦D)V )
∂ , D
[ ]
∂ij d ij
subject to
≽ 0 , N ≥ j > i = 1... N −1
d ij 1
(1291)
D ∈ EDM N
∂ ∈ S N h
7.16 For nonzero Y ∈ S N h and some open interval of t∈R (3.2.3.0.2,D.2.3)
d 2
dt 2 ‖(D + tY ) − H‖2 F = 2 trY T Y > 0
7.17 If that H given has negative entries, then the technique of solution presented here
becomes invalid. Projection of H on K (1206) prior to application of this proposed
technique, as explained in7.0.1, is incorrect.