Column Generation for Extended Formulations

Working in an extended variable space allows one to develop tight reformulations for mixed integer programs (MIP). However, the size of the extended formulation grows rapidly too large for a direct treatment by a MIP-solver. When the extended formulation stems from a decomposition principle, as typical in practice, a column generation procedure that mimics that for the Dantzig-Wolfe reformulation can be applied to the extended formulation. Pricing subproblem solutions are expressed in the variables of the extended formulation and added to the current restricted version of the extended formulation along with the subproblem constraints that are active for the subproblem solution. Such “column-and-row generation” procedure is reviewed with the goal to analyse the approachʼs potential benefits compared to a standard column generation approach. Numerical experiments highlight a key observation: lifting pricing problem solutions in the space of the extended formulation permits their recombination into new subproblem solutions and results in faster convergence.