ColumnGeneration

Column generation is a mathematical optimization technique used to solve large-scale linear programming problems. It is particularly useful when the number of variables in the problem is extremely large, making direct solution methods computationally infeasible. The method was first introduced by George Dantzig and Delbert Ray Fulkerson in 1954.

In column generation, the problem is decomposed into a master problem and one or more subproblems. The master problem is a restricted version of the original problem, containing only a subset of the variables. The subproblems are used to generate new variables (columns) that can improve the solution of the master problem. This process is repeated iteratively until no further improvements can be made.

The master problem is typically solved using the simplex method or an interior-point method. The subproblems are usually solved using dynamic programming or other specialized algorithms. The dual variables from the master problem are used to define the objective function of the subproblems, ensuring that only variables that can improve the solution are generated.

Column generation has been successfully applied to a wide range of problems, including vehicle routing, crew scheduling, and network design. It is particularly effective when the problem exhibits a natural decomposition, allowing for efficient solution of the subproblems. The method can significantly reduce the computational burden of solving large-scale linear programming problems, making it an important tool in operations research and optimization.