🤖 AI Summary
To address the poor scalability of conventional real sum-of-squares (RSOS) methods—which rely on large-scale semidefinite programming (SDP) for computing lower bounds on the minimum of real homogeneous polynomials over the unit sphere—this paper proposes a purely spectral (non-SDP) convergent hierarchy. The key innovation is the first rigorous reduction of real spherical optimization to Hermitian optimization, enabling the construction of a sequence of lower bounds via minimal eigenvalue computations alone, within the Hermitian sum-of-squares (HSOS) framework. This approach naturally extends to estimating the spectral norm of real tensors, thereby opening a new pathway for spectral methods in general constrained real optimization. Numerical experiments and asymptotic analysis demonstrate substantial improvements over RSOS and other baseline methods; moreover, the framework yields a computable, convergent hierarchy for the spectral norm.
📝 Abstract
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.
