Symmetric tensor networks for generative modeling and constrained combinatorial optimization

Abstract

Constrained combinatorial optimization problems abound in industry, from portfolio optimization to logistics. One of the major roadblocks in solving these problems is the presence of non-trivial hard constraints which limit the valid search space. In some heuristic solvers, these are typically addressed by introducing certain Lagrange multipliers in the cost function, by relaxing them in some way, or worse yet, by generating many samples and only keeping valid ones, which leads to very expensive and inefficient searches. In this work, we encode arbitrary integer-valued equality constraints of the form $A \vec{x} = \vec{b}$ , directly into U (1) symmetric tensor networks (TNs) and leverage their applicability as quantum-inspired generative models to assist in the search of solutions to combinatorial optimization problems within the generator-enhanced optimization framework of Alcazar et al (2021 arXiv: 2101.06250 ). This allows us to exploit the generalization capabilities of TN generative models while constraining them so that they only output valid samples. Our constrained TN generative model efficiently captures the constraints by reducing number of parameters and computational costs. We find that at tasks with constraints given by arbitrary equalities, symmetric matrix product states outperform their standard unconstrained counterparts at finding novel and better solutions to combinatorial optimization problems.

Keywords

• generative learning
• combinatorial optimization
• tensor network
• constraints
• U(1) symmetry