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arXiv:2301.07244 · Journal of the Physical Society of Japan

Efficient Correlation-based Discretization of Continuous Variables for Annealing Machines

Yuki Furue, Makiko Konoshima, Hirotaka Tamura, Jun Ohkubo

Published Mar 23, 2023• 1 citation
arXivHTMLPDF

Abstract

Annealing machines specialized for combinatorial optimization problems have been developed, and some companies offer services to use those machines. Such specialized machines can only handle binary variables, and their input format is the quadratic unconstrained binary optimization (QUBO) formulation. Therefore, discretization is necessary to solve problems with continuous variables. However, there is a severe constraint on the number of binary variables with such machines. Although the simple binary expansion in the previous research requires many binary variables, we need to reduce the number of such variables in the QUBO formulation due to the constraint. We propose a discretization method that involves using correlations of continuous variables. We numerically show that the proposed method reduces the number of necessary binary variables in the QUBO formulation without a significant loss in prediction accuracy.

DOI
10.7566/jpsj.92.044802

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Annealing machines specialized for combinatorial optimization problems have been developed, and some companies offer services to use those machines. Such specialized machines can only handle binary variables, and their input format is the quadratic unconstrained binary optimization (QUBO) formulation. Therefore, discretization is necessary to solve problems with continuous variables. However, there is a severe constraint on the number of binary variables with such machines. Although the simple binary expansion in the previous research requires many binary variables, we need to reduce the number of such variables in the QUBO formulation due to the constraint. We propose a discretization method that involves using correlations of continuous variables. We numerically show that the proposed method reduces the number of necessary binary variables in the QUBO formulation without a significant loss in prediction accuracy.
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Quantum annealing and related combinatorial optimization methods typically accept quadratic unconstrained binary optimization (QUBO) problems as input, whereas many practical models include constraints and continuous variables. Standard QUBO conversions discretize continuous variables, increasing the binary dimension and often making feasible low-energy states harder to sample. We develop a Lagrange-multiplier method for a separable class of mixed-binary quadratic programs (MBQPs) without discretizing continuous variables. The method is based on a finite-temperature free-energy formulation and uses QUBO sampling for the binary sector. At fixed Lagrange multipliers, the continuous sector is integrated out analytically and enters only the multiplier update, leaving a QUBO over the original binary variables. We evaluate the method on the continuous relaxation of the quadratic ppp-median problem. Compared with a penalty-based QUBO formulation, it generates feasible solutions more reliably. At an appropriate inverse temperature, its conditional relative error is comparable to that of local search for small instances and often lower for the larger tested instances. In the time-to-target comparisons, the proposed method maintains high target-reached rates for stringent fixed-accuracy targets. For the larger instances within the tested range, it also reaches the prescribed targets faster than a commercial mixed-integer optimization solver.
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We propose a spin-variable reduction method for Ising machines to handle linear equality constraints in a combinatorial optimization problem. Ising machines including quantum-annealing machines can effectively solve combinatorial optimization problems. They are designed to find the lowest-energy solution of a quadratic unconstrained binary optimization (QUBO), which is mapped from the combinatorial optimization problem. The proposed method reduces the number of binary variables to formulate the QUBO compared to the conventional penalty method. We demonstrate a sufficient condition to obtain the optimum of the combinatorial optimization problem in the spin-variable reduction method and its general applicability. We apply it to typical combinatorial optimization problems, such as the graph κ-partitioning problem and the quadratic assignment problem. Experiments using simulated-annealing and quantum-annealing based Ising machines demonstrate that the spin-variable reduction method outperforms the penalty method. The proposed method extends the application of Ising machines to larger-size combinatorial optimization problems with linear equality constraints.
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Ising machines, including quantum annealing machines, are promising next-generation computers for combinatorial optimization problems. However, due to hardware limitations, most Ising-type hardware can only solve objective functions expressed in linear or quadratic terms of binary variables. Therefore, problems with higher-order terms require an order-reduction process, which increases the number of variables and constraints and may degrade solution quality. In this study, we evaluate the effectiveness of directly solving such problems without order reduction by using a high-performance simulated annealing-based optimization solver capable of handling polynomial unconstrained binary optimization (PUBO) formulations. We compare its performance against a conventional quadratic unconstrained binary optimization (QUBO) solver on the same hardware platform. As benchmarks, we use the low autocorrelation binary sequence (LABS) problem and the vehicle routing problem with distance balancing, both of which naturally include higher-order interactions. Results show that the PUBO solver consistently achieves superior solution quality and stability compared to its QUBO counterpart, while maintaining comparable computational time and requiring no order-reduction compilation indicating potential advantages of directly handling higher-order terms in practical optimization problems.