# Benders decomposition

**Benders decomposition** (or **Benders' decomposition**) is a technique in mathematical programming that allows the solution of very large linear programming problems that have a special block structure. This block structure often occurs in applications such as stochastic programming as the uncertainty is usually represented with scenarios. The technique is named after Jacques F. Benders.

The strategy behind Benders decomposition can be summarized as *divide-and-conquer*. That is, in Benders decomposition, the variables of the original problem are divided into two subsets so that a first-stage master problem is solved over the first set of variables, and the values for the second set of variables are determined in a second-stage subproblem for a given first-stage solution. If the subproblem determines that the fixed first-stage decisions are in fact infeasible, then so-called *Benders cuts* are generated and added to the master problem, which is then re-solved until no cuts can be generated. Since Benders decomposition adds new *constraints* as it progresses towards a solution, the approach is called "*row* generation". In contrast, Dantzig–Wolfe decomposition uses "*column* generation".

## See also

- FortSP solver uses Benders decomposition for solving stochastic programming problems

## References

- Benders, J. F. (Sept. 1962), "Partitioning procedures for solving mixed-variables programming problems",
*Numerische Mathematik*4(3): 238–252. - Lasdon, Leon S. (2002),
*Optimization Theory for Large Systems*(reprint of the 1970 Macmillan ed.), Mineola, New York: Dover Publications, pp. xiii+523, MR 1888251.