Cons-training Tensor Networks: Embedding and Optimization Over Discrete Linear Constraints

TL;DR

This paper introduces constrained tensor networks called MPS that incorporate discrete linear constraints, improving optimization efficiency and scalability for complex combinatorial problems.

Contribution

The paper develops a new family of tensor networks with a canonical form for efficiently modeling and optimizing over discrete linear constraints.

Findings

Successfully applied to quadratic knapsack problem with superior results

Provides a scalable approach for constrained combinatorial optimization

Introduces quantum region concept for flexible constraint modeling

Abstract

In this study, we introduce a novel family of tensor networks, termed constrained matrix product states (MPS), designed to incorporate exactly arbitrary discrete linear constraints, including inequalities, into sparse block structures. These tensor networks are particularly tailored for modeling distributions with support strictly over the feasible space, offering benefits such as reducing the search space in optimization problems, alleviating overfitting, improving training efficiency, and decreasing model size. Central to our approach is the concept of a quantum region, an extension of quantum numbers traditionally used in U(1) symmetric tensor networks, adapted to capture any linear constraint, including the unconstrained scenario. We further develop a novel canonical form for these new MPS, which allow for the merging and factorization of tensor blocks according to quantum region fusion rules and permit optimal truncation schemes. Utilizing this canonical form, we apply an unsupervised training strategy to optimize arbitrary objective functions subject to discrete linear constraints. Our method's efficacy is demonstrated by solving the quadratic knapsack problem, achieving superior performance compared to a leading nonlinear integer programming solver. Additionally, we analyze the complexity and scalability of our approach, demonstrating its potential in addressing complex constrained combinatorial optimization problems.