2023/10/27 by Benjamin Lovitz, Nathaniel Johnston, Lovitz, Benjamin +1 · 1 citation
Computer Science · Mathematics · #Advanced Optimization Algorithms Research #Algebraic Geometry (math.AG) #Data Structures and Algorithms (cs.DS) #FOS: Computer and information sciences #FOS: Mathematics #FOS: Physical sciences #Matrix Theory and Algorithms #Optimization and Control (math.OC) #Quantum Physics (quant-ph) #Tensor decomposition and applications
paper · pdf · doi:10.48550/arxiv.2310.17827
openalex publication_date 2023/10/27 · openalex created_date 2023/11/01 · openalex updated_date 2026/07/28
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.