Generating Benders Cuts for a General Class of Integer Programming Problems

Abstract

This paper proposes a method of generating valid integer Benders cuts for a general class of integer programming problems. A generic valid Benders cut in disjunctive form is presented first, as a basis for the subsequent derivations of simple valid cuts. Under a qualification condition, a simple valid Benders cut in linear form can be identified. A cut generation problem is formulated to elicit it. The simple valid Benders cut is further generalized to a minimally relaxed Benders cut, based on which a complete Benders decomposition algorithm is given, and its finite convergency to optimality is proved. The proposed algorithm provides a way of applying the Benders decomposition strategy to solve integer programs. The computational results show that using the Benders algorithm for integer programs to exploit the problem structures can reduce the solving time more and more as the problem size increases.