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Column generation is often used to solve models with stronger linear-programming relaxations. From the dual standpoint, column-generation processes can be viewed as cutting plane algorithms. In this paper, we present conditions under which it is possible to restrict the dual space, but still preserve the strength of the primal model and recover an optimal primal solution. We derive a family of dual cuts that are valid for the space of dual optimal solutions of the one-dimensional cutting-stock problem. These cuts correspond to extra columns in the primal model. Inserting a polynomial number of cuts of this family in the problem formulation at initialization time restricts the dual space during the entire column-generation process. Computational experiments show that this idea helps, reducing substantially the number of columns generated, the number of degenerate iterations, and the computational time.

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Dual-feasible functions are valuable tools that can be used to compute both lower bounds for different combinatorial problems and valid inequalities for integer programs. Several families of functions have been used in the literature. Some of them were defined explicitly, and others not. One of the main objectives of this paper is to survey these functions, and to state results concerning their quality. We clearly identify dominant subsets of functions, i.e. those which may lead to better bounds or stronger cuts. We also describe different frameworks that can be used to create dualfeasible functions. With these frameworks, one can get a dominant function based on other ones. Two new families of dual-feasible functions obtained by applying these methods are proposed in this paper.We also performed a computational comparison on the relative strength of the functions presented in this paper for deriving lower bounds for the bin-packing problem and valid cutting planes for the pattern minimization problem. Extensive experiments on instances generated using methods described in the literature are reported. In many cases, the lower bounds are improved, and the linear relaxations are strengthened.