2024/06/13 by Martin Plávala, Plávala, Martin, Laurens T. Ligthart +3 · 1 citation
Computer Science · Mathematics · #Advanced Optimization Algorithms Research #FOS: Mathematics #FOS: Physical sciences #Matrix Theory and Algorithms #Optimization and Control (math.OC) #Quantum Physics (quant-ph)
paper · pdf · doi:10.48550/arxiv.2406.09506
openalex publication_date 2024/06/13 · openalex created_date 2024/06/18 · openalex updated_date 2026/07/28
We construct a convergent family of outer approximations for the problem of optimizing polynomial functions over convex bodies subject to polynomial constraints. This is achieved by generalizing the polarization hierarchy, which has previously been introduced for the study of polynomial optimization problems over state spaces of C^*-algebras, to convex cones in finite dimensions. If the convex bodies can be characterized by linear or semidefinite programs, then the same is true for our hierarchy. Convergence is proven by relating the problem to a certain de Finetti theorem for general probabilistic theories, which are studied as possible generalizations of quantum mechanics. We apply the method to the problem of nonnegative matrix factorization, and in particular to the nested rectangles problem. A numerical implementation of the third level of the hierarchy is shown to give rise to a very tight approximation for this problem.