Polynomial-size formulations and relaxations for the quadratic multiple knapsack problem

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.