Alternative Uses of the Open Approach
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
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.
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 MINLPTerminate; else ! 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.