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 Ax=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. 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.

v2. Improved version: restructured the paper; corrected complexity analysis and put more details of the encoding algorithm; added new, and improved some existing figures for clarity; added a few extra refs. v3. Shortened captions, added new table with outline of algorithm

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• Cons-training Tensor Networks: Embedding and Optimization Over Discrete Linear Constraints
• TensorKrowch: Smooth integration of tensor networks in machine learning
• Systematic and Efficient Construction of Quadratic Unconstrained Binary Optimization Forms for High-order and Dense Interactions
• Quick design of feasible tensor networks for constrained combinatorial optimization
• Variational matrix product states for combinatorial optimization