v2009.01.01 - Convex Optimization

5.4. EDM DEFINITION 361
ten affine equality constraints in all on a Gram matrix G∈ S 6 (820). Let’s
realize this as a convex feasibility problem (with constraints written in the
same order) also assuming 0 geometric center (819):
find
D∈S 6 h
−V DV 1 2 ∈ S6
subject to tr ( )
D(e i e T j + e j e T i ) 1 2 = l
2
ij , j−1 = (i = 1... 6) mod 6
tr ( − 1V DV (A 2 i + A T i ) 2) 1 = cos ϕi , i = 1, 2, 3
tr(− 1 2 V DV ) = 1
−V DV ≽ 0 (832)
where, for A i ∈ R 6×6 (827)
A 1 = (e 1 − e 6 )(e 3 − e 4 ) T /(l 61 l 34 )
A 2 = (e 2 − e 1 )(e 4 − e 5 ) T /(l 12 l 45 )
A 3 = (e 3 − e 2 )(e 5 − e 6 ) T /(l 23 l 56 )
(833)
and where the first constraint on length-square l 2 ij can be equivalently written
as a constraint on the Gram matrix −V DV 1 via (829). We show how to
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numerically solve such a problem by alternating projection inE.10.2.1.1.
Barvinok’s Proposition 2.9.3.0.1 asserts existence of a list, corresponding
to Gram matrix G solving this feasibility problem, whose affine dimension
(5.7.1.1) does not exceed 3 because the convex feasible set is bounded by
the third constraint tr(− 1 V DV ) = 1 (823).
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5.4.2.2.3 Example. Kissing-number of sphere packing.
Two nonoverlapping Euclidean balls are said to kiss if they touch. An
elementary geometrical problem can be posed: Given hyperspheres, each
having the same diameter 1, how many hyperspheres can simultaneously
kiss one central hypersphere? [345] The noncentral hyperspheres are allowed,
but not required, to kiss.
As posed, the problem seeks the maximal number of spheres K kissing
a central sphere in a particular dimension. The total number of spheres is
N = K + 1. In one dimension the answer to this kissing problem is 2. In two
dimensions, 6. (Figure 7)
The question was presented in three dimensions to Isaac Newton in the
context of celestial mechanics, and became controversy with David Gregory
on the campus of Cambridge University in 1694. Newton correctly identified