Procedure GMP::SolverSession::AddBendersOptimalityCut(solverSession, GMP, solution, local, purgeable)


The procedure GMP::SolverSession::AddBendersOptimalityCut generates an optimality cut for a Benders’ master problem using the (dual) solution of a Benders’ subproblem. The Benders’ master problem must be a MIP problem. The cut is typically added as a lazy constraint in a callback during the MIP branch-and-cut search. This procedure is typically used in a Benders’ decomposition algorithm in which a single master MIP problem is solved.

     solverSession,    ! (input) a solver session
     GMP,              ! (input) a generated mathematical program
     solution,         ! (input) a solution
     [local],          ! (optional, default 0) a scalar binary expression
     [purgeable]       ! (optional, default 0) a scalar binary expression



An element in the set AllSolverSessions representing a solver session for the Benders’ master problem.


An element in the set AllGeneratedMathematicalPrograms representing a Benders’ subproblem.


An integer scalar reference to a solution of GMP2.


A scalar binary value to indicate whether the cut is valid for the local problem (i.e. the problem corresponding to the current node in the solution process and all its descendant nodes) only (value 1) or for the global problem (value 0).


A scalar binary value to indicate whether the solver is allowed to purge the cut if it deems it ineffective. If the value is 1, then it is allowed.

Return Value

The procedure returns 1 on success, or 0 otherwise.


  • The generated mathematical program corresponding to the solverSession should have been created using the function GMP::Benders::CreateMasterProblem.

  • The GMP should have been created using the function GMP::Benders::CreateSubProblem.

  • The solution of the Benders’ subproblem (represented by GMP) is used to generate an optimality cut for the Benders’ master problem (represented by solverSession). More precise, the shadow prices of the constraints and the reduced costs of the variables in the Benders’ subproblem are used.

  • See Benders’ Decomposition - Textbook Algorithm of the Language Reference for more information about the Benders’ decomposition algorithm in which a single master MIP problem is solved.


In the example below we solve only one Benders’ master problem (which is a MIP). During the solve, whenever the solver finds an integer (incumbent) solution we want to run a callback for lazy constraints. Therefore we install a callback for it.

myGMP := GMP::Instance::Generated( MP );

gmpM := GMP::Benders::CreateMasterProblem( myGMP, AllIntegerVariables,
                                           'BendersMasterProblem', 0, 0 );

gmpS := GMP::Benders::CreateSubProblem( myGMP, masterGMP, 'BendersSubProblem',
                                        0, 0 );

GMP::Instance::SetCallbackAddLazyConstraint( gmpM, 'LazyCallback' );

GMP::Instance::Solve( gmpM );

The callback procedure LazyCallback has one argument, namely ThisSession which is an element parameter with range AllSolverSessions. Inside the callback procedure we solve the Benders’ subproblem. We assume that the Benders’ subproblem is always feasible. The program status of the subproblem is stored in the element parameter ProgramStatus with range AllSolutionStates. Note that the subproblem is updated before it is solved.

! Get MIP incumbent solution.
GMP::Solution::RetrieveFromSolverSession( ThisSession, 1 );
GMP::Solution::SendToModel( gmpM, 1 );

GMP::Benders::UpdateSubProblem( gmpS, gmpM, 1, round : 1 );

GMP::Instance::Solve( gmpS );

ProgramStatus := GMP::Solution::GetProgramStatus( gmpS, 1 ) ;
if ( ProgramStatus = 'Optimal' ) then
    GMP::SolverSession::AddBendersOptimalityCut( ThisSession, gmpF, 1 );

In this example we skipped the check for optimality of the Benders’ decomposition algorithm.