Notes on Dantzig-Wolfe decomposition and column generation

Here X is a vector of variables, all lying in the set S. The cost vector is denoted c. A is a matrix, b a vector, and together they form the constraints on the variables. When S is the set of real numbers, the mathematical model is said to be a linear program (LP). If S is a binary set or the set of integers, the problem is an integer program (IP). If some variables in X belong to the set of real numbers and others to the set of binaries or integers, the model is said to be a mixed integer program (MIP). Let X ∈ R : x ≥ 0, x ∈ X in the above formulation and denote this the primal problem. The dual model is found by transposing the primal problem: