# Automatic Benders’ Decomposition

Benders’ decomposition is an approach to solve complicated mathematical programming problems by splitting them into two, and thereby simplifying the solution process by (repeatedly) solving one master problem and one subproblem. If the problem contains integer variables then typically they become part of the master problem while the continuous variables become part of the subproblem. The classic approach of the Benders’ decomposition algorithm solves an alternating sequence of master problems and subproblems. Benders’ decomposition is mostly used for solving difficult MIP problems and stochastic programming problems.

We have developed a generic Benders’ decomposition module in AIMMS that is easy to use.

The Benders’ decomposition module in AIMMS is implemented as an open algorithm using GMP functionality. It offers many features. Besides the classic approach the module contains a modern variant that solves one master (MIP) problem using solver callbacks. (See also Paul Rubin’s blog post Benders Decomposition Then and Now.)

To use Benders’ decomposition, first install the system module `GMPBendersDecomposition`.

To solve a MIP problem using the classic approach you can use the following:

```1! First we must generate the GMP for our Math Program.
2myGMP := GMP::Instance::Generate( myMP ) ;
3
4! The second argument defines the master problem variables.
5GMPBenders::DoBendersDecomposition( myGMP, AllIntegerVariables, 'Classic' );
```

The third argument (`'Classic'`) of the above procedure specifies the algorithm that will be used. Other possible values are `'Modern'`, `'TwoPhaseClassic'` and `'TwoPhaseModern'`. The two-phase algorithm solves a relaxed problem in the first phase and the original MIP problem in the second phase (using either the classic or modern approach).

The current implementation of Benders’ decomposition in AIMMS has some limitations. It cannot be used to solve nonlinear problems. Also, the current implementation does not support multiple subproblems which could be efficient in case the subproblem has a block diagonal structure. This implies that the current implementation cannot be used to solve (two stage) stochastic programming problems with a subproblem for each scenario.

CPLEX nowadays provides its own Benders Decomposition. See Solve with Benders Decomposition in CPLEX.