具有晶体对称性的三角多项式优化与避免图的谱界
Optimization of trigonometric polynomials with crystallographic symmetry and spectral bounds for set avoiding graphs
AI总结：
本文提出利用晶体对称性优化三角多项式的方法，用于计算避免图的谱界。
摘要：
三角多项式通常定义在整数格上。我们考虑更广泛的加权和根格，具有晶体对称性。本文提出了一种新的方法来最小化在相关反射群下不变的三角多项式。不变性假设允许我们将目标函数用广义切比雪夫多项式来重写。新的目标函数定义在紧致基本半代数集上，因此可以利用丰富的多项式优化理论。我们提出了一种计算最小值的算法：基于Hol-Scherer正态展开定理，我们在切比雪夫基上对目标函数施加矩阵和平方条件。平方的次数被加权定义，由根系决定。增加次数会得到一个收敛的Lasserre型下界层次结构。这建立了三角和多项式优化之间的桥梁，使我们能够与现有技术进行比较。在欧几里得空间中，避免图的色数通过最优着色定义。可以通过最小化三角多项式来计算其谱界。如果要避免的集合具有晶体对称性，我们的方法有自然的应用。具体来说，我们首次为对称多面体的边界计算了谱界。对于几种情况，问题有简化形式，我们可以给出精确谱界的解析证明。在其他情况下，我们通过数值方法证明了精确性。
英文摘要：
Trigonometric polynomials are usually defined on the lattice of integers.We consider the larger class of weight and root lattices with crystallographic symmetry.This article gives a new approach to minimize trigonometric polynomials, which are invariant under the associated reflection group.The invariance assumption allows us to rewrite the objective function in terms of generalized Chebyshev polynomials. The new objective function is defined on a compact basic semi-algebraic set, so that we can benefit from the rich theory of polynomial optimization.We present an algorithm to compute the minimum: Based on the Hol-Scherer Positivstellensatz, we impose matrix-sums of squares conditions on the objective function in the Chebyshev basis.The degree of the sums of squares is weighted, defined by the root system. Increasing the degree yields a converging Lasserre-type hierarchy of lower bounds.This builds a bridge between trigonometric and polynomial optimization, allowing us to compare with existing techniques.The chromatic number of a set avoiding graph in the Euclidean space is defined through an optimal coloring.It can be computed via a spectral bound by minimizing a trigonometric polynomial. If the to be avoided set has crystallographic symmetry, our method has a natural application.Specifically, we compute spectral bounds for the first time for boundaries of symmetric polytopes.For several cases, the problem has such a simplified form that we can give analytical proofs for sharp spectral bounds.In other cases, we certify the sharpness numerically.
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