v2009.01.01 - Convex Optimization

5.4. EDM DEFINITION 363
spheres:
Then the kissing number
minimize − tr W V T
D∈S N
N DV N
subject to D 1j = 1, j = 2... N
(834)
D ij ≥ 1, 2 ≤ i < j = 3... N
D ∈ EDM N
K = N max − 1 (835)
is found from the maximal number of spheres N that solve this semidefinite
program in a given affine dimension r . Matrix W can be interpreted as the
direction of search through the positive semidefinite cone S N−1
+ for a rank-r
optimal solution −VN TD⋆ V N ; it is constant, in this program, determined by
a method disclosed in4.4.1. In two dimensions,
⎡
⎤
4 1 2 −1 −1 1
1 4 −1 −1 2 1
W =
2 −1 4 1 1 −1
1
⎢ −1 −1 1 4 1 2
(836)
⎥6
⎣ −1 2 1 1 4 −1 ⎦
1 1 −1 2 −1 4
In three dimensions,
⎡
⎤
9 1 −2 −1 3 −1 −1 1 2 1 −2 1
1 9 3 −1 −1 1 1 −2 1 2 −1 −1
−2 3 9 1 2 −1 −1 2 −1 −1 1 2
−1 −1 1 9 1 −1 1 −1 3 2 −1 1
3 −1 2 1 9 1 1 −1 −1 −1 1 −1
W =
−1 1 −1 −1 1 9 2 −1 2 −1 2 3
1
−1 1 −1 1 1 2 9 3 −1 1 −2 −1
12
1 −2 2 −1 −1 −1 3 9 2 −1 1 1
2 1 −1 3 −1 2 −1 2 9 −1 1 −1
⎢ 1 2 −1 2 −1 −1 1 −1 −1 9 3 1
⎥
⎣ −2 −1 1 −1 1 2 −2 1 1 3 9 −1 ⎦
1 −1 2 1 −1 3 −1 1 −1 1 −1 9
(837)
A four-dimensional solution has rational direction matrix as well. Here is an
optimal point list 5.11 in Matlab output format:
5.11 An optimal five-dimensional point list is known: The answer was known at least 175