Optimality of spherical codes via exact semidefinite programming bounds

H Cohn, D de Laat, N Leijenhorst - arxiv preprint arxiv:2403.16874, 2024‏ - arxiv.org
We show that the spectral embeddings of all known triangle-free strongly regular graphs are
optimal spherical codes (the new cases are $56 $ points in $20 $ dimensions, $50 $ points …

Optimality and uniqueness of the root system

D de Laat, NM Leijenhorst, WHH Keizer - arxiv preprint arxiv:2404.18794, 2024‏ - arxiv.org
We prove that the $ D_4 $ root system (the set of vertices of the regular $24 $-cell) is the
unique optimal kissing configuration in $\mathbb R^ 4$, and is an optimal spherical code …

Solving clustered low-rank semidefinite programs arising from polynomial optimization

N Leijenhorst, D de Laat - Mathematical Programming Computation, 2024‏ - Springer
We study a primal-dual interior point method specialized to clustered low-rank semidefinite
programs requiring high precision numerics, which arise from certain multivariate …

The Lasserre hierarchy for equiangular lines with a fixed angle

D de Laat, FC Machado, WM Keizer - arxiv preprint arxiv:2211.16471, 2022‏ - arxiv.org
We compute the second and third levels of the Lasserre hierarchy for the spherical finite
distance problem. A connection is used between invariants in representations of the …

Optimal measures for multivariate geometric potentials

D Bilyk, D Ferizović, A Glazyrin, RW Matzke… - arxiv preprint arxiv …, 2023‏ - arxiv.org
We study measures and point configurations optimizing energies based on multivariate
potentials. The emphasis is put on potentials defined by geometric characteristics of sets of …

Dual certificates and efficient rational sum-of-squares decompositions for polynomial optimization over compact sets

MM Davis, D Papp - SIAM Journal on Optimization, 2022‏ - SIAM
We study the problem of computing weighted sum-of-squares (WSOS) certificates for
positive polynomials over a compact semialgebraic set. Building on the theory of interior …

Verifying feasibility of degenerate semidefinite programs

V Kolmogorov, S Naldi, J Zapata - arxiv preprint arxiv:2405.13625, 2024‏ - arxiv.org
This paper deals with the algorithmic aspects of solving feasibility problems of semidefinite
programming (SDP), aka linear matrix inequalities (LMI). Since in some SDP instances all …

The smallest mono-unstable convex polyhedron with point masses has 8 faces and 11 vertices

D Papp, K Regős, G Domokos, S Bozóki - European Journal of Operational …, 2023‏ - Elsevier
In the study of monostatic polyhedra, initiated by John H. Conway in 1966, the main question
is to construct such an object with the minimal number of faces and vertices. By …

Additive and Multiplicative Coinvariant Spaces of Weyl Groups in the Light of Harmonics and Graded Transfer

S Debus, T Metzlaff - arxiv preprint arxiv:2412.17099, 2024‏ - arxiv.org
A finite group with an integral representation has two induced canonical actions, one on
polynomials and one on Laurent polynomials. Knowledge about the invariants is in either …

Optimization of trigonometric polynomials with crystallographic symmetry and spectral bounds for set avoiding graphs

E Hubert, T Metzlaff, P Moustrou, C Riener - Mathematical Programming, 2024‏ - Springer
We provide a new approach to the optimization of trigonometric polynomials with
crystallographic symmetry. This approach widens the bridge between trigonometric and …