v2007.09.17 - Convex Optimization

5.4. EDM DEFINITION 309spheres:Then the kissing numberminimize − tr W V TD∈S NN DV Nsubject to D 1j = 1, j = 2... N(745)D ij ≥ 1, 2 ≤ i < j = 3... ND ∈ EDM NK = N max − 1 (746)is found from the maximal number of spheres N that solve this semidefiniteprogram in a given affine dimension r . Matrix W can be interpreted as thedirection of search through the positive semidefinite cone S N−1+ for a rank-roptimal solution −VN TD⋆ V N ; it is constant, in this program, determined bya method disclosed in4.4.1. In two dimensions,⎡⎤4 1 2 −1 −1 11 4 −1 −1 2 1W =2 −1 4 1 1 −11⎢ −1 −1 1 4 1 2(747)⎥6⎣ −1 2 1 1 4 −1 ⎦1 1 −1 2 −1 4In three dimensions,⎡⎤9 1 −2 −1 3 −1 −1 1 2 1 −2 11 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 13 −1 2 1 9 1 1 −1 −1 −1 1 −1W =−1 1 −1 −1 1 9 2 −1 2 −1 2 31−1 1 −1 1 1 2 9 3 −1 1 −2 −1121 −2 2 −1 −1 −1 3 9 2 −1 1 12 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(748)A four-dimensional solution has rational direction matrix as well. Here is anoptimal 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