On symmetry adapted bases in trigonometric optimization

TL;DR

This work addresses the global minimization of multivariate trigonometric polynomials that are invariant under a finite group action. It advances a sums-of-squares relaxation framework by solving SDPs on Hermitian Toeplitz matrices and then exploiting crystallographic symmetry via a symmetry adapted basis to achieve block-diagonalization, substantially reducing computational cost. Key contributions include a constructive method to obtain symmetry adapted bases through isotypic decomposition, explicit procedures for projection and basis construction, and a detailed A2/S3 example that demonstrates practical efficiency and numerical fidelity. The approach broadens SOS-based trigonometric optimization to general symmetric settings and provides a scalable pathway for certification via SDP in high-symmetry problems.

Abstract

The problem of computing the global minimum of a trigonometric polynomial is computationally hard. We address this problem for the case, where the polynomial is invariant under the exponential action of a finite group. The strategy is to follow an established relaxation strategy in order to obtain a converging hierarchy of lower bounds. Those bounds are obtained by numerically solving semi-definite programs (SDPs) on the cone of positive semi-definite Hermitian Toeplitz matrices, which is outlined in the book of Dumitrescu [Dum07]. To exploit the invariance, we show that the group has an induced action on the Toeplitz matrices and prove that the feasible region of the SDP can be restricted to the invariant matrices, whilst retaining the same solution. Then we construct a symmetry adapted basis tailored to this group action, which allows us to block-diagonalize invariant matrices and thus reduce the computational complexity to solve the SDP. The approach is in its generality novel for trigonometric optimization and complements the one that was proposed as a poster at the ISSAC 2022 conference [HMMR22] and later extended to [HMMR24]. In the previous work, we first used the invariance of the trigonometric polynomial to obtain a classical polynomial optimization problem on the orbit space and subsequently relaxed the problem to an SDP. Now, we first make the relaxation and then exploit invariance. Partial results of this article have been presented as a poster at the ISSAC 2023 conference [Met23].

Figures (5)

• Figure 1: The subsets $\{0\} = \Omega_0 \subseteq \Omega_1 \subseteq \ldots \subseteq \Omega_4$ of the hexagonal lattice in the plane (specifically the weight lattice of the root system $\mathrm{A}_{2}$) are contained in scaled copies of the Voronoï cell of the dual lattice.
• Figure 2: The root system $\mathrm{A}_{2}$ in $\mathbb{R}^3/\langle [1,1,1]^t \rangle$.
• Figure 3: The root system $\mathrm{B}_{2}$ in $\mathbb{R}^2$.
• Figure 4: The root system $\mathrm{G}_{2}$ in $\mathbb{R}^3/\langle [1,1,1]^t \rangle$.
• Figure 5: The root system $\mathrm{C}_{2}$ in $\mathbb{R}^2$.

Theorems & Definitions (16)

• Lemma 2.1
• proof
• Example 2.2: Using the notation from \ref{['appendix_root_systems']}
• Lemma 2.3
• proof
• Proposition 2.4
• proof
• Example 3.1
• Proposition 3.2
• proof