[论文解读] A Tutorial on Formulating QUBO Models.

Fred Glover, Gary Kochenberger|arXiv (Cornell University)|Nov 13, 2018

Operations Management Techniques被引用 20

一句话总结

本文系统性地介绍了如何使用简单直接的重构技术，将组合优化问题建模为无约束二次二值优化（QUBO）模型。通过将经典优化问题转化为QUBO形式，该方法可借助专用的QUBO求解器实现高效求解，从而确立QUBO作为解决各类工业与政府优化挑战的统一框架。

ABSTRACT

The field of Combinatorial Optimization (CO) is one of the most important areas in the general field of optimization, with important applications found in every industry, including both the private and public sectors. It is also one of the most active research areas pursued by the research communities of Operations Research, Computer Science, and Analytics as they work to design and test new methods for solving real world CO problems. In recent years, we have discovered that a mathematical formulation known as QUBO, an acronym for a Quadratic Unconstrained Binary Optimization problem, can embrace a large variety of important CO problems found in industry and government. Through special reformulation techniques that are easy to apply, the power of QUBO solvers can be used to efficiently solve many important problems once they are put into the QUBO framework. This two-step process of first re-casting an original model into the form of a QUBO model and then solving it with appropriate software enables the QUBO model to become a unifying framework for combinatorial optimization. The alternative path that results for effectively modeling and solving many important problems is a new development in the field of combinatorial optimization. The materials provided in subsequent sections illustrate the process of reformulating important optimization problems as QUBO models through a series of explicit examples. Collectively these examples highlight the application breadth of the QUBO model. We do not focus here on modeling a given problem in the form classically adopted in the optimization community. Rather, our examples disclose the methods that can be used to recast a given problem, perhaps existing in a classical mathematical form, into an equivalent QUBO model.

研究动机与目标

• 展示大量组合优化问题均可被重构为QUBO形式。
• 提供一个统一的建模框架，使QUBO求解器能够应用于各种现实世界问题。
• 将关注点从经典优化建模转向QUBO重构，作为一种强大且替代性的方法。
• 通过具体、分步的示例，说明QUBO的广泛应用性。
• 赋予研究人员和实践者将现有模型转换为QUBO兼容形式的工具。

提出的方法

• 本文采用重构技术，将经典组合优化问题的数学模型转化为等价的QUBO公式。
• 引入二值变量以表示离散选择，从而以二次形式表达逻辑与组合约束。
• 通过惩罚项将约束嵌入目标函数，以在QUBO公式中保持可行性。
• 通过数学变换确保原始问题与最终QUBO模型之间的等价性。
• 使用示例演示该过程，展示不同类型问题（如集合划分、图划分）如何映射到QUBO。
• 该方法避免依赖传统优化建模格式，转而专注于QUBO特定的重构方法。

实验结果

研究问题

• RQ1 经典组合优化问题如何被系统性地转化为QUBO模型？
• RQ2 哪些重构技术在将约束和目标函数转换为二次二值形式时最为有效？
• RQ3 QUBO在多大程度上可作为各类优化问题的统一框架？
• RQ4 确保原始模型与其QUBO重构模型之间等价性的关键步骤与原则是什么？
• RQ5 从业者如何在无量子启发优化先验知识的情况下，将QUBO建模应用于现实世界问题？

主要发现

• QUBO框架能够表示广泛的组合优化问题，包括工业与政府领域的问题。
• 将问题重构为QUBO形式后，可借助专用QUBO求解器实现高效求解，提升问题求解的可扩展性。
• 该转换过程系统且易于理解，依赖于适用于多种问题类型的明确定义的技术。
• 该方法保持了问题的等价性，确保QUBO模型的解对原始问题同样有效。
• 该教程表明，QUBO建模是经典优化建模方法的一种可行且强大的替代方案。
• 该方法凸显了QUBO作为组合优化中统一范式所具备的潜力。