This paper addresses the Quadratic Multiple Constraints Variable-Sized Bin Packing Problem (QMC-VSBPP), a challenging combinatorial optimization problem that generalizes the classical bin packing problem by incorporating multiple capacity dimensions, heterogeneous bin types, and quadratic interaction costs between items. We propose two complementary methods that advance the current state-of-the-art. First, a linearized mathematical model is introduced to eliminate quadratic terms, enabling the use of exact solvers such as Gurobi to compute strong lower bounds, reported here for the first time for this problem. Second, we develop RKO-ACO, a continuous-domain Ant Colony Optimization algorithm within the Random-Key Optimizer framework, enhanced with adaptive Q-learning parameter control and efficient local search. Extensive computational experiments on benchmark instances show that the proposed linearized model produces significantly tighter lower bounds than the original quadratic model, while RKO-ACO consistently matches or improves upon all best-known solutions in the literature, establishing new upper bounds for large-scale instances. These results provide new reference values for future studies and demonstrate the effectiveness of evolutionary and random-key approaches for solving complex quadratic packing problems. Source code and data available at https://github.com/nataliaalves03/RKO-ACO

翻译：本文针对二次多约束变尺寸装箱问题（QMC-VSBPP）展开研究，该问题是一类具有挑战性的组合优化问题，通过引入多个容量维度、异质箱子类型以及物品间的二次交互成本，推广了经典装箱问题。我们提出两种互补方法来提升现有最优水平。首先，引入线性化数学模型以消除二次项，从而能够使用Gurobi等精确求解器计算强下界——这是首次针对该问题报告此类下界。其次，我们开发了RKO-ACO算法，这是一种基于随机密钥优化器框架的连续域蚁群优化算法，并集成了自适应Q学习参数控制与高效局部搜索。在基准测试实例上的大量计算实验表明，所提出的线性化模型比原始二次模型能产生显著更紧凑的下界，而RKO-ACO算法一致地匹配或改进了文献中所有已知最优解，为大规模实例建立了新的上界。这些结果为后续研究提供了新的参考值，并证明了进化方法与随机密钥途径在求解复杂二次装箱问题中的有效性。源代码和数据可在https://github.com/nataliaalves03/RKO-ACO获取。