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 x ⃗ = 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.
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 x ⃗ = 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.
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) 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.
Abstract Devising an efficient exploration of the search space is one of the key challenges in the design of combinatorial optimization algorithms. Here, we introduce the Generator-Enhanced Optimization (GEO) strategy: a framework that leverages any generative model (classical, quantum, or quantum-inspired) to solve optimization problems. We focus on a quantum-inspired version of GEO relying on tensor-network Born machines, and referred to hereafter as TN-GEO. To illustrate our results, we run these benchmarks in the context of the canonical cardinality-constrained portfolio optimization problem by constructing instances from the S&P 500 and several other financial stock indexes, and demonstrate how the generalization capabilities of these quantum-inspired generative models can provide real value in the context of an industrial application. We also comprehensively compare state-of-the-art algorithms and show that TN-GEO is among the best; a remarkable outcome given the solvers used in the comparison have been fine-tuned for decades in this real-world industrial application. Also, a promising step toward a practical advantage with quantum-inspired models and, subsequently, with quantum generative models
Quantum computers are expected to enable fast solving of large-scale combinatorial optimization problems. However, their limitations in fidelity and the number of qubits prevent them from handling real-world problems. Recently, a quantum-inspired solver using tensor networks has been proposed, which works on classical computers. Particularly, tensor networks have been applied to constrained combinatorial optimization problems for practical applications. By preparing a specific tensor network to sample states that satisfy constraints, feasible solutions can be searched for without the method of penalty functions. Previous studies have been based on profound physics, such as U(1) gauge schemes and high-dimensional lattice models. In this study, we devise to design feasible tensor networks using elementary mathematics without such a specific knowledge. One approach is to construct tensor networks with nilpotent-matrix manipulation. The second is to algebraically determine tensor parameters. We showed mathematically that such feasible tensor networks can be constructed to accommodate various types of constraints. For the principle verification, we numerically constructed a feasible tensor network for facility location problem, to find much faster construction than conventional methods. Then, by performing imaginary time evolution, feasible solutions were always obtained, ultimately leading to the optimal solution.
Optimization is a crucial task in various industries such as logistics, aviation, manufacturing, chemical, pharmaceutical, and insurance, where finding the best solution to a problem can result in significant cost savings and increased efficiency. Tensor networks (TNs) have gained prominence in recent years in modeling classical systems with quantum-inspired approaches. More recently, TN generative-enhanced optimization (TN-GEO) has been proposed as a strategy which uses generative modeling to efficiently sample valid solutions with respect to certain constraints of optimization problems. Moreover, it has been shown that symmetric TNs (STNs) can encode certain constraints of optimization problems, thus aiding in their solution process. In this work, we investigate the applicability of TN- and STN-GEO to an industry relevant problem class, a multi-knapsack problem, in which each object must be assigned to an available knapsack. We detail a prescription for practitioners to use the TN-and STN-GEO methodology and study its scaling behavior and dependence on its hyper-parameters. We benchmark 60 different problem instances and find that TN-GEO and STN-GEO produce results of similar quality to simulated annealing.
Combinatorial optimization is one of the key candidates in the race for practical quantum advantage. In this work, we introduce a new family of quantum-enhanced optimizers and demonstrate how quantum machine learning models known as quantum generative models can find lower minima than those found by means of stand-alone state-of-the-art classical solvers. We present two new quantum-enhanced optimization strategies. The first scheme leverages data points evaluated during the optimization search from any quantum or classical optimizer. In this scheme, we show how our quantum generative model boosts the performance of classical solvers in hard-to-solve instances where the classical solver is not capable of making progress as a stand-alone strategy. The second quantum optimization strategy works as a stand-alone solver. Here we show its superior performance when the goal is to find the best minimum within the least number of cost function evaluations. Under this setting, we benchmark our quantum-enhanced optimization strategies against several solvers, including Bayesian optimizers which are known to be one of the best competing solvers in such tasks. To illustrate our findings, these benchmarks are performed in the context of the portfolio optimization problem by constructing instances from the S&P 500 stock market index. We show that our quantum-inspired generative models based on tensor networks generalize to unseen candidates with lower cost function values than any of the candidates seen by the classical solvers. This is the first demonstration of the generalization capabilities of quantum generative models that brings real value in the context of an industrial-scale application.
We introduce a new framework that leverages machine learning models known as generative models to solve optimization problems. Our Generator-Enhanced Optimization (GEO) strategy is flexible to adopt any generative model, from quantum to quantum-inspired or classical, such as Generative Adversarial Networks, Variational Autoencoders, or Quantum Circuit Born Machines, to name a few. Here, we focus on a quantum-inspired version of GEO relying on tensor-network Born machines and referred to hereafter as TN-GEO. We present two prominent strategies for using TN-GEO. The first uses data points previously evaluated by any quantum or classical optimizer, and we show how TN-GEO improves the performance of the classical solver as a standalone strategy in hard-to-solve instances. The second strategy uses TN-GEO as a standalone solver, i.e., when no previous observations are available. Here, we show its superior performance when the goal is to find the best minimum given a fixed budget for the number of function calls. This might be ideal in situations where the cost function evaluation can be very expensive. To illustrate our results, we run these benchmarks in the context of the portfolio optimization problem by constructing instances from the S\&P 500 and several other financial stock indexes. We also comprehensively compare state-of-the-art algorithms in a generalized version of the portfolio optimization problem. The results show that TN-GEO is among the best compared to these state-of-the-art algorithms; a remarkable outcome given the solvers used in the comparison have been fine-tuned for decades in this real-world industrial application. We see this as an important step toward a practical advantage with quantum-inspired models and, subsequently, with quantum generative models.