Abstract
Dual-feasible functions have been used to compute fast lower bounds and valid inequalities for integer linear optimization problems. However, almost all the functions proposed in the literature are defined only for positive arguments, which restricts considerably their applicability. The characteristics and properties of dual-feasible functions with general domains remain mostly unknown. In this paper, we show that extending these functions to negative arguments raises many issues. We explore these functions in depth with a focus on maximal functions, i.e. the family of non-dominated functions. The knowledge of these properties is fundamental to derive good families of general maximal dual-feasible functions that might lead to strong cuts for integer linear optimization problems and strong lower bounds for combinatorial optimization problems with knapsack constraints.
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