In colloquial terms, column generation is a way of beginning with a small, manageable part of a problem (specifically, a few of the variables), solving that part, analyzing that partial solution to determine the next part of the problem (specifically, one or more variables) to add to the model, and then resolving the enlarged model. Column-wise modeling repeats that process until it achieves a satisfactory solution to the whole of the problem.

In formal terms, column generation is a way of solving a linear programming problem that adds columns (corresponding to constrained variables) during the pricing phase of the simplex method of solving the problem. In gross terms, generating a column in the primal simplex formulation of a linear programming problem corresponds to adding a constraint in its dual formulation. In the dual formulation of a given linear programming problem, you might think of column generation as a cutting plane method.

In that context, many researchers have observed that column generation is a very powerful technique for solving a wide range of industrial problems to optimality or to near optimality. Ford and Fulkerson, for example, suggested column generation in the context of a multi-commodity network flow problem as early as 1958 in the journal of Management Science. By 1960, Dantzig and Wolfe had adapted it to linear programming problems with a decomposable structure. Gilmore and Gomory then demonstrated its effectiveness in a cutting stock problem. More recently, vehicle routing, crew scheduling, and other integer-constrained problems have motivated further research into column generation.

Column generation rests on the fact that in the simplex method, the solver does not need access to all the variables of the problem simultaneously. In fact, a solver can begin work with only the basis (a particular subset of the constrained variables) and then use reduced cost to determine which other variables to access as needed.