基于情景簇的增强型渐进对冲算法求解两阶段随机二次背包问题

中文摘要 AI 辅助

本文提出了两阶段随机二次背包问题（TSSQKP），其中利润和权重均存在不确定性。为克服由随机性、二元性和二次结构相结合所带来的计算困难，我们提出了一种将渐进对冲算法的改进版本与基于聚类的场景缩减相结合的求解框架。增强型渐进对冲算法（EPHA）引入了舍入过程以及由振荡和停滞模式检测驱动的动态罚函数更新策略。同时，我们提出了一种基于场景间机会成本距离的聚类框架，并由此推导出中位数下界（MLB）、簇下界（CLB）和簇上界（CUB）。为进一步提高计算效率，EPHA还被用于启发式地求解簇子问题；在这种情况下，所得上界被称为估计簇上界（ECUB），相应的间隙被解释为估计最优性间隙。在800个生成的TSSQKP实例上进行的计算实验表明，在相当的计算条件下，所提出的框架比CPLEX产生了更小的间隙。在基于EPHA的聚类框架中，与CLB、EPHA和MLB相关的平均估计间隙分别为1.65%、1.88%和2.41%，而CPLEX的平均最优性间隙为35.86%。MLB计算速度最快，平均执行时间为420.88秒，而EPHA在解质量和计算工作量之间提供了有利的折衷，将簇界相关的平均计算时间减少了约52.7%，同时平均估计间隙仅增加了0.23个百分点。

英文摘要

This paper introduces the two-stage stochastic quadratic knapsack problem (TSSQKP) with uncertainty in profits and weights. To overcome the computational difficulties arising from the combined stochastic, binary, and quadratic structure, we propose a solution framework that integrates an adaptation of the progressive hedging algorithm with clustering-based scenario reduction. The enhanced progressive hedging algorithm (EPHA) incorporates a rounding procedure and a dynamic penalty update strategy driven by the detection of oscillation and stagnation patterns. In parallel, we propose a clustering framework based on opportunity-cost distances between scenarios, from which we derive the medoid lower bound (MLB), the cluster lower bound (CLB) and the cluster upper bound (CUB). To further improve computational efficiency, EPHA is also used to solve the cluster subproblems heuristically; in this case, the resulting upper bound is referred to as the estimated cluster upper bound (ECUB), and the corresponding gaps are interpreted as estimated optimality gaps. Computational experiments on 800 generated TSSQKP instances show that the proposed framework yields smaller gaps than CPLEX under comparable computational conditions. In the EPHA-based clustering framework, the average estimated gaps associated with the CLB, EPHA and MLB 1.65%, 1.88% and 2.41% respectively, compared with an average optimality gap of 35.86% for CPLEX. MLB is fastest to compute, with an average execution time of 420.88s, while EPHA provides a favourable compromise between solution quality and computational effort, reducing the average computational time associated with the cluster bounds by approximately 52.7%, while increasing the average estimated gap by only 0.23 percentage points.

发表机构

• GERAD, CIRRELT & Department of Analytics, Operations and Information Technology, Université du Québec à Montréal（蒙特利尔大学 GERAD、CIRRELT 与分析、运营与信息技术系）
• GERAD, CIRRELT & Department of Operations and Decision Systems, Université Laval（魁北克大学拉瓦尔分校 GERAD、CIRRELT 与运营与决策系统系）
• GERAD, CRM & Department of Analytics, Operations and Information Technology, Université du Québec à Montréal（蒙特利尔大学 GERAD、CRM 与分析、运营与信息技术系）

机构由 AI 辅助整理，请以论文原文为准。

补充信息

备注

42 pages, 2 figures