Alternative Uses of the Open Approach

Customize algorithm

Using the open outer approximation approach for solving MINLP models it is possible to add to the existing procedures or write alternative procedures to meet the needs of the final user. For instance, a user evaluating the performance of the algorithm may want to add certain performance measurements and print statements to the existing code. Some less trivial examples of modifications are provided in the next few paragraphs.

Solve more NLPs

Practical experience has shown that it is sometimes difficult to get a feasible solution to the initial relaxed NLP model. Based on the particular application, the user may specify how multiple starting values can be found, and then modify the algorithm to solve multiple NLPs to get a feasible and/or a better solution. While doing so, it is also possible to specify how the algorithm should switch between different solvers (using the predefined AIMMS identifier CurrentSolver). Such extensions could then also be applied to the NLP subproblem inside the While statement.

Retain integer solutions

It is possible to activate a MIP callback procedure whenever the MIP solver finds an integer solution. Even though these intermediate solutions are not optimal, the user may want to save the integer portion of these solutions for later evaluation. Once the main algorithm has terminated, all these integer solutions can be retrieved and evaluated by solving the corresponding nonlinear subproblem. In some instances, one of these extra solutions may be a better solution to the original MINLP model than the one produced by the main algorithm.

Adjust penalties

Setting the penalties for the deviations of the linear approximation constraints in the master MIP subproblem is a delicate manner, and has an effect on the solution quality when the nonlinear subproblems are nonconvex. The user can consider several problem-dependent strategies to adjust the penalty values, and implement them inside the basic AOA algorithm.

Example of modified procedure

The following procedure is a variant of the termination procedure provided in the previous section. Assuming that the two parameters that refer to the previous and current NLP objective function values have been properly set in the procedure that solves the NLP subproblem, then termination is invoked whenever there is insufficient progress between two subsequent NLP solutions, or between the objective values of the master MIP problem and the current NLP subproblem. The third termination criterion is the number of iterations reaching its maximum.

return when ( MINLPAlgorithmHasFinished );

if    (not MINLPSolutionImprovement( NLPCurrentObjectiveValue,
                                     NLPPreviousObjectiveValue ))
   or (not MINLPSolutionImprovement( GMP::Solution::GetObjective(GMINLP, SolNumb),
                                     NLPCurrentObjectiveValue ))
   or ( IterationCounter = IterationMax ) then


    ! Prepare for next iteration

    IterationCount += 1 ;
    GMP::Solution::SetIterationCount( GMINLP, SolNumb, IterationCount ) ;
    GMP::Instance::AddIntegerEliminationRows( GMIP, SolNumb, EliminationCount ) ;
    EliminationCount += 1 ;
endif ;


The above paragraphs indicate just a few of the ways in which you can alter the basic implementation of the outer approximation algorithm in AIMMS. Of course, it is not necessary to develop your own variant. Whenever you need to solve a MINLP model using the AOA algorithm, you can simply call the basic implementation described in the previous section. As soon as you can see improved ways to solve a particular model, you can apply your own ideas by modifying the procedures as you see fit.