可扩展的惩罚权重确定方法用于近似求解器上的约束优化
Scalable Determination of Penalization Weights for Constrained Optimizations on Approximate Solvers
AI总结：
本文提出了一种可扩展的方法，用于确定约束优化中惩罚权重，适用于基于吉布斯求解器的多项式复杂度问题，实验显示在大规模实例上具有鲁棒性和显著加速。
摘要：
二次无约束二元优化（QUBO）为各种计算问题提供问题表述，可通过专用QUBO求解器解决，这些求解器可以基于经典或量子计算。解决约束组合优化问题的常见方法是在QUBO公式中强制约束，通过添加惩罚项实现。惩罚引入了一个额外的超参数，显著影响求解器的效果：目标项与惩罚项之间的相对权重。我们开发了一种预计算策略，用于确定惩罚权重，为吉布斯求解器提供可证明的保证，并对广泛的问题类别具有多项式复杂度。在多样化的问题和求解器架构上的实验，包括福岛数字退火器上的大规模实例，显示了稳健的性能和数量级的加速。
英文摘要：
Quadratic unconstrained binary optimization (QUBO) provides problem formulations for various computational problems that can be solved with dedicated QUBO solvers, based on classical or quantum computation. A common approach to constrained combinatorial optimization problems is to enforce the constraints in the QUBO formulation by adding penalization terms. Penalization introduces an additional hyperparameter that significantly affects the solver's efficacy: the relative weight between the objective terms and the penalization terms. Here we study the problem of a priori determining suitable penalization weights from a foundational and practical perspective. We formally establish that finding its optimal value for Gibbs solvers is $\mathsf{NP}$-hard even approximately and $\#\mathsf{P}$-hard exactly. This serves us as a motivation to develop a widely applicable heuristic strategy for determining suitable penalization weights, with provable guarantees for Gibbs solvers, and polynomial complexity for broad problem classes. Experiments across diverse problems and solver architectures, including large-scale instances on Fujitsu's Digital Annealer, show robust performance and order-of-magnitude speedups over existing heuristics. We further extend the strategy to quantum approximate optimization and demonstrate its performance on Quantinuum's H2-2 trapped-ion processor.
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