Higher-order cover cuts from zero–one knapsack constraints augmented by two-sided bounding inequalities

Abstract

Extending our work on second-order cover cuts [F. Glover, H.D. Sherali, Second-order cover cuts, Mathematical Programming (ISSN: 0025-5610 1436-4646) (2007), doi:10.1007/s10107-007-0098-4. (Online)], we introduce a new class of higher-order cover cuts that are derived from the implications of a knapsack constraint in concert with supplementary two-sided inequalities that bound the sums of sets of variables. The new cuts can be appreciably stronger than the second-order cuts, which in turn dominate the classical knapsack cover inequalities. The process of generating these cuts makes it possible to sequentially utilize the second-order cuts by embedding them in systems that define the inequalities from which the higher-order cover cuts are derived. We characterize properties of these cuts, design specialized procedures to generate them, and establish associated dominance relationships. These results are used to devise an algorithm that generates all non-dominated higher-order cover cuts, and, in particular, to formulate and solve suitable separation problems for deriving a higher-order cut that deletes a given fractional solution to an underlying continuous relaxation. We also discuss a lifting procedure for further tightening any generated cut, and establish its polynomial-time operation for unit-coefficient cuts. A numerical example is presented that illustrates these procedures and the relative strength of the generated non-redundant, non-dominated higher-order cuts, all of which turn out to be facet-defining for this example. Some preliminary computational results are also presented to demonstrate the efficacy of these cuts in comparison with lifted minimal cover inequalities for the underlying knapsack polytope.