Composition and tensor train structure in polynomial optimization

TL;DR

This paper introduces two novel moment-SOS hierarchies that leverage composition and tensor train structures in polynomial optimization, enabling efficient bounds computation for large-scale problems.

Contribution

The authors develop two new hierarchies exploiting problem structure, improving scalability and applicability in diverse fields like control, Markov chains, and neural networks.

Findings

Hierarchies can handle problems with hundreds to a thousand variables.

Numerical experiments show certified bounds are efficiently computed.

Methods are versatile across applications like quantum control and neural networks.

Abstract

We study polynomial optimization problems whose objective has a composition or tensor train structure. These polynomials can be evaluated as a sequence of maps, giving rise to intermediate variables (``states'') of dimension lower than the ambient dimension. Structures like these arise naturally in dynamical systems, Markov chains, and neural networks. We develop two moment-SOS (sums of squares) hierarchies that exploit this composition structure in different ways. The first one, termed state-lifting chordal, is based on the correlative sparsity graph of the problem. The second one, termed state-lifting push-forward, encodes the structure at the level of the measures directly. Numerical experiments demonstrate that the proposed methods can compute certified bounds for problems with hundreds or even a thousand variables. To illustrate the versatility of the hierarchies we apply them to Markov chain optimization, quantum optimal control, and neural networks.