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arXiv 2511.00520math.OCmath.CO
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结合基于梯度启发式方法的割平面法

Cutting plane methods with gradient-based heuristics

Hòa T. Bùi, Alberto De Marchi

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8 pages + references and appendix, 10 figures, 3 algorithms, 1 table
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AI总结：

针对非线性离散优化问题，提出结合割平面法全局收敛性与投影梯度法局部探索能力的混合方法，通过在更接近最优解集处构造割平面，显著提升了解的质量与计算效率。

摘要：

割平面法，特别是外部逼近法，是求解非线性离散优化问题的一种成熟方法，它无需放松决策变量的整数性。尽管在理论上很强大，但其计算性能可能极不稳定。近期研究表明，在不可行点向可行集的投影处构造割平面，能显著提升割平面法的性能。受此启发，我们研究了在更接近最优解集的可行点处构造割平面是否能进一步增强割平面法的有效性。我们提出了一种混合方法，将割平面法的全局收敛保证与一阶优化技术的局部探索能力相结合。具体而言，我们使用投影梯度法作为启发式策略，以识别解空间中具有潜力的区域，并生成更紧致、更具信息量的割平面。我们重点关注具有凸可微目标函数的二元优化问题，其中投影操作可通过混合整数线性规划高效计算。通过在更接近最优解集的点处构造割平面并排除非最优区域，该算法实现了对可行域的更好逼近和更快的收敛速度。数值实验证实，我们的方法在不同的求解器配置下均提升了解的质量和计算效率。该框架为进一步扩展到更一般的离散领域提供了灵活的基础，并为非线性离散优化工具箱提供了一种极具前景的启发式方法。

英文摘要：

Cutting plane methods, particularly outer approximation, are a well-established approach for solving nonlinear discrete optimization problems without relaxing the integrality of decision variables. While powerful in theory, their computational performance can be highly variable. Recent research has shown that constructing cutting planes at the projection of infeasible points onto the feasible set can significantly improve the performance of cutting plane approaches. Motivated by this, we examine whether constructing cuts at feasible points closer to the optimal solution set could further enhance the effectiveness of cutting plane methods. We propose a hybrid method that combines the global convergence guarantees of cutting plane methods with the local exploration capabilities of first-order optimization techniques. Specifically, we use projected gradient methods as a heuristic to identify promising regions of the solution space and generate tighter, more informative cuts. We focus on binary optimization problems with convex differentiable objective functions, where projection operations can be efficiently computed via mixed-integer linear programming. By constructing cuts at points closer to the optimal solution set and eliminating non-optimal regions, the algorithm achieves better approximation of the feasible region and faster convergence. Numerical experiments confirm that our approach improves both the quality of the solution and computational efficiency across different solver configurations. This framework provides a flexible foundation for further extensions to more general discrete domains and offers a promising heuristic to the toolkit for nonlinear discrete optimization.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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