In this talk, we give a primal-dual interior point method specialized to clustered low-rank semidefinite programs, which arise from multivariate polynomial (matrix) programs. This extends the work of Simmons-Duffin [J. High Energ. Phys. 1506, no. 174 (2015)] from univariate to multivariate polynomial matrix programs, and to more general clustered low-rank semidefinite programs.
We study the interplay of sampling (Löfberg and Parrilo [43rd IEEE CDC (2004)]) and symmetry reduction as well as a method to obtain numerically good bases and sample points. We apply this to the computation of three-point bounds for the kissing number problem, for which we show a significant speedup (easily by a factor 20 for previously computed bounds). We also use the extra stability due to the chosen bases and samples to perform computations for the binary sphere packing problem.