Abstract The Quadratic Multiple Knapsack Problem generalizes, simultaneously, two well-known combinatorial optimization problems that have been intensively studied in the literature: the (single) Quadratic Knapsack Problem and the Multiple Knapsack Problem. The only exact algorithm for its solution uses a formulation based on an exponential-size number of variables, that is solved via a Branch-and-Price algorithm. This work studies polynomial-size formulations and upper bounds. We derive linear models from classical reformulations of 0-1 quadratic programs and analyze theoretical properties and dominances among them. We define surrogate and Lagrangian relaxations, and we compare the effectiveness of the Lagrangian relaxation when applied to a quadratic formulation and to a Level 1 reformulation linearization that leads to a decomposable structure. The proposed methods are evaluated through extensive computational experiments.

中文翻译：

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二次多重背包问题的多项式大小公式和松弛

摘要 二次多背包问题同时概括了文献中已经深入研究的两个众所周知的组合优化问题：（单）二次背包问题和多背包问题。其解决方案的唯一精确算法使用基于指数大小变量数量的公式，该公式通过分支和价格算法求解。这项工作研究多项式大小的公式和上限。我们从 0-1 二次规划的经典重构中推导出线性模型，并分析它们的理论特性和优势。我们定义了代理和拉格朗日松弛，并比较了拉格朗日松弛在应用于二次公式和导致可分解结构的 1 级重新公式线性化时的有效性。