AI 中文总结
本文针对双层双重防御攻击者模型，提出基于对偶化的精确锚定方法与混合启发式算法，验证了其最优性并在多数配置上优于已发表结果。
AI 中文摘要
双层阻断模型常采用元启发式算法求解，但由于测试规模下无法获得精确最优解，其解的质量难以评估。本文针对双层双重防御攻击者模型（BDAM）开展研究，该模型结合了节点阻断、边破坏和带容量的供应支持，此前我们采用混合元启发式算法求解。首先，当弧长与防御者的二元决策呈线性关系时，BDAM的主导攻击者路径项可通过下层对偶化转化为精确的单级公式，所得混合整数规划（MILP）可对早期基准的18种三行和五行配置验证最优性，其中17种在10秒内完成，而15×30网格的强未验证 incumbent 仍可扩展。我们已发表的平均值较该前沿低3.55%，且差距随规模扩大而增大。其次，供应规则如同基于单条攻击者最短路径定价的任何规则，在存在平局时定义不明确，而本文提出的威胁走廊公式在构造上具有平局不变性。平局打破操作对目标函数的影响小于10^(-3)，但实际供应成本变化可达0.83单位，因此该缺陷是由目标权重导致而非不存在。最后，MILP-DA将精确锚定与走廊解码及可行性修复相结合，与已发表结果（双方均未重新实现）相比，在18种已验证配置中的13种以及所有18种更大规模配置上表现更优，所有失败均发生在三行网格上。
英文摘要
Bi-level interdiction models are frequently solved by metaheuristics whose solution quality cannot be assessed, because exact optima are unavailable at the scales tested. We supply them for the bi-level dual-defense attacker model (BDAM), which couples node interdiction, edge destruction and capacitated supply support, and which we previously solved by a hybrid metaheuristic. First, BDAM's dominant attacker-path term admits an exact single-level reformulation by lower-level dualization, a reduction available whenever arc lengths are linear in the defender's binary decisions; the resulting mixed-integer program certifies optimality on all eighteen three-row and five-row configurations of our earlier benchmark, seventeen in under ten seconds, with strong uncertified incumbents out to 15*30 grids. Our published averages sit 3.55% below that frontier, and the gap widens with scale. Second, the supply rule, like any rule priced on a single attacker shortest path, is ill-posed under ties, and our threat-corridor formulation is tie-invariant by construction. The tie-break moves the objective by under 10^(-3) but the realized supply cost by up to 0.83 units, so the defect is suppressed by the objective weight rather than absent. Third, MILP-DA pairs the exact anchor with the corridor decode and a feasibility repair; compared against our published figures with no re-implementation on either side, it wins on thirteen of eighteen certified configurations and on all eighteen larger ones, every loss falling on a three-row grid.
CommentsSubmitted to the European Journal of Operational Research
