Exploiting nested inequalities and surrogate constraints

The exploitation of nested inequalities and surrogate constraints as originally proposed in Glover [Glover, F., 1965. A multiphase-dual algorithm for the zero-one integer programming problem. Operations Research 13, 879-919; Glover, F., 1971. Flows in arborescences. Management Science 17, 568-586] has been specialized to multidimensional knapsack problems in Osorio et al. [Osorio, M.A., Glover, F., Hammer, P., 2002. Cutting and surrogate constraint analysis for improved multidimensional knapsack solutions. Annals of Operations Research 117, 71-93]. We show how this specialized exploitation can be strengthened to give better results. This outcome results by a series of observations based on surrogate constraint duality and properties of nested inequalities. The consequences of these observations are illustrated by numerical examples to provide insights into uses of surrogate constraints and nested inequalities that can be useful in a variety of problem settings.