A Simple Perspective on Simultaneous Column and Row Generation

To solve a linear programming problem with many variables and constraints, a simultaneous column and row generation algorithm can be used. Such an algorithm iteratively solves a restricted version of a linear programming problem including only a subset of the variables and constraints. In each iteration, it might simultaneously add both a new variable and constraint, as opposed to non-simultaneous column and row generation algorithms that only add either a new variable or constraint. In this paper, a simple perspective is provided to indicate the relevance of simultaneous column and row generation and to make it more accessible. Particular emphasis is put on an optimality condition for linear programming which is perhaps a particular source of confusion in this context. It is argued that optimality can be achieved using non-simultaneous column and row generation algorithms, while an example is provided illustrating the potential computational gains of using simultaneous column and row generation algorithms.