Solving Mixed Integer Programming Problems Using Benders Decomposition Algorithm

In mixed integer programming (MIP) problems, the Benders decomposition algorithm serves as a highly efficient solution method. This algorithm decomposes large-scale optimization problems into smaller, more manageable subproblems - typically separating integer variables from continuous variables. The master problem handles integer decisions while subproblems verify feasibility and generate optimality cuts. Through iterative computations, the algorithm constructs Benders cuts that progressively tighten the solution space, ultimately converging to the global optimum. A key advantage lies in its ability to significantly reduce computational complexity by avoiding excessive binary variable computations during the solution process. Implementation typically involves:

1. Formulating the master problem with integer variables and temporary objective bounds

2. Solving subproblems for given integer solutions to generate feasibility/optimality cuts

3. Iteratively adding Benders cuts to the master problem until convergence

From a coding perspective, the algorithm can be implemented using optimization libraries like Gurobi or CPLEX, where callback functions manage cut generation. The typical workflow involves initializing the master problem, solving it to obtain integer solutions, then solving the dual subproblem to produce cutting planes. This approach proves particularly valuable when dealing with problems featuring complicating variables that, when fixed, make the remaining problem more tractable. Therefore, when confronting challenging mixed integer programming problems, Benders decomposition algorithm presents a robust computational framework worth considering.