v2009.01.01 - Convex Optimization

524 CHAPTER 7. PROXIMITY PROBLEMS
constraint to the objective:
minimize −〈V (D − 2Y )V , I〉 − w〈VN TDV N , W 〉
D , Y
[ ]
dij y ij
subject to
≽ 0 , N ≥ j > i = 1... N −1
y ij h 2 ij
(1282)
Y ∈ S N h
D ∈ EDM N
where w (≈ 10) is a positive scalar just large enough to make 〈V T N DV N , W 〉
vanish to within some numerical precision, and where direction matrix W is
an optimal solution to semidefinite program (1581a)
minimize −〈VN T W
D⋆ V N , W 〉
subject to 0 ≼ W ≼ I
trW = N − 1
(1283)
one of which is known in closed form. Semidefinite programs (1282) and
(1283) are iterated until convergence in the sense defined on page 278. This
iteration is not a projection method. Convex problem (1282) is neither a
relaxation of unidimensional scaling problem (1281); instead, problem (1282)
is a convex equivalent to (1281) at convergence of the iteration.
Jan de Leeuw provided us with some test data
⎡
H =
⎢
⎣
0.000000 5.235301 5.499274 6.404294 6.486829 6.263265
5.235301 0.000000 3.208028 5.840931 3.559010 5.353489
5.499274 3.208028 0.000000 5.679550 4.020339 5.239842
6.404294 5.840931 5.679550 0.000000 4.862884 4.543120
6.486829 3.559010 4.020339 4.862884 0.000000 4.618718
6.263265 5.353489 5.239842 4.543120 4.618718 0.000000
and a globally optimal solution
⎤
⎥
⎦
(1284)
X ⋆ = [ −4.981494 −2.121026 −1.038738 4.555130 0.764096 2.822032 ]
= [ x ⋆ 1 x ⋆ 2 x ⋆ 3 x ⋆ 4 x ⋆ 5 x ⋆ 6 ]
(1285)
found by searching 6! local minima of (1213) [89]. By iterating convex
problems (1282) and (1283) about twenty times (initial W = 0) we find the