We introduce a convergent hierarchy of lower bounds on the minimum value of a real form over the unit sphere. The main practical advantage of our hierarchy over the real sum-of-squares (RSOS) 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 RSOS. In practice, this allows us to compute bounds on much larger forms than are computationally feasible for RSOS. Our hierarchy outperforms previous alternatives to RSOS, both asymptotically and in numerical experiments. We obtain our hierarchy by proving a reduction from real optimization on the sphere to Hermitian optimization on the sphere, and invoking the Hermitian sum-of-squares (HSOS) hierarchy. This opens the door to using other Hermitian optimization techniques for real optimization, and gives a path towards developing spectral hierarchies for more general constrained real optimization problems. To this end, we use our techniques to develop a hierarchy of eigencomputations for computing the real tensor spectral norm.

翻译：我们提出了一种收敛的层次结构，用于计算实形式在单位球面上的最小值下界。与实数和平方（RSOS）层次结构相比，我们提出的层次结构的主要实际优势在于：每一级的下界仅需通过最小特征值计算获得，而RSOS的每一级则需要求解完整的半定规划（SDP）。在实践中，这使得我们能够计算比RSOS在计算上可行的大得多的形式的下界。无论是在渐近性能还是在数值实验中，我们的层次结构均优于以往RSOS的替代方案。我们通过证明从球面上的实优化到球面上的埃尔米特优化的约简，并调用埃尔米特和平方（HSOS）层次结构，从而得到我们的层次结构。这为利用其他埃尔米特优化技术进行实优化开辟了道路，并为发展更一般的约束实优化问题的谱层次结构提供了路径。为此，我们利用我们的技术开发了一种用于计算实张量谱范数的特征计算层次结构。