We introduce a convergent hierarchy of lower bounds on the minimum value of a real homogeneous polynomial over the sphere. The main practical advantage of our hierarchy over the sum-of-squares (SOS) hierarchy is that the lower bound at each level of our hierarchy is obtained by a minimum eigenvalue computation, as opposed to the full semidefinite program (SDP) required at each level of SOS. In practice, this allows us to go to much higher levels than are computationally feasible for the SOS hierarchy. For both hierarchies, the underlying space at the $k$-th level is the set of homogeneous polynomials of degree $2k$. We prove that our hierarchy converges as $O(1/k)$ in the level $k$, matching the best-known convergence of the SOS hierarchy when the number of variables $n$ is less than the half-degree $d$ (the best-known convergence of SOS when $n \geq d$ is $O(1/k^2)$). More generally, we introduce a convergent hierarchy of minimum eigenvalue computations for minimizing the inner product between a real tensor and an element of the spherical Segre-Veronese variety, with similar convergence guarantees. As examples, we obtain hierarchies for computing the (real) tensor spectral norm, and for minimizing biquadratic forms over the sphere. Hierarchies of eigencomputations for more general constrained polynomial optimization problems are discussed.

翻译：我们提出了一组收敛的下界层级，用于计算实齐次多项式在球面上的最小值。与平方和（SOS）层级相比，我们的层级在实际应用中的主要优势在于：每一层下界可通过最小特征值计算获得，而无需像SOS层级那样在每个水平求解完整的半定规划（SDP）。实践中，这使得我们能够达到远高于SOS层级计算可行性的水平。对于两个层级，第k层的基础空间均为次数为2k的齐次多项式集合。我们证明该层级在水平k处以O(1/k)的速度收敛，当变量数n小于半次数d时（当n≥d时SOS的最佳已知收敛速度为O(1/k^2)），这与SOS层级的最佳已知收敛速度相匹配。更一般地，我们引入了一组收敛的特征计算层级，用于最小化实张量与球面Segre-Veronese簇元素之间的内积，并具有类似的收敛保证。作为示例，我们获得了计算（实）张量谱范数的层级，以及最小化球面上双二次型的层级。文中还讨论了更一般的约束多项式优化问题的特征计算层级。