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 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 formulation 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 Optimization 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 formulation, 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 metaheuristic 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，该问题是一个具有挑战性的组合优化问题，它在经典装箱问题的基础上引入了多维容量约束、异构箱型以及物品间的二次交互成本。我们提出了两种互补的方法，以推进当前的研究前沿。首先，引入了一种线性化的数学模型，以消除二次项，从而能够使用Gurobi等精确求解器计算强下界——这是该问题首次报道此类下界。其次，我们开发了RKO-ACO算法，这是一种在随机密钥优化框架下构建的连续域蚁群优化算法，通过自适应Q学习参数控制和高效局部搜索进行了增强。在基准实例上进行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产生了显著更紧的下界，而RKO-ACO算法则持续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。这些结果为未来研究提供了新的参考值，并证明了进化和随机密钥元启发式方法在解决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。