Algorithm in words
The algorithm solves an alternating sequence of mixed-integer linear models and nonlinear models.
First, the entire model is solved as a nonlinear program with all the integer variables relaxed as continuous variables between their bounds.
Then a linearization is carried out around the optimal solution, and the resulting constraints are added to the linear constraints that are already present. This new linear model is referred to as the master MIP model.
The master MIP problem is solved as an mixed-integer linear program.
The integer part of the resulting optimal solution is then temporarily fixed, and the original MINLP model with fixed integer variables is solved as a nonlinear subproblem.
Again, a linearization around the optimal solution is constructed and the new linear constraints are added to the master MIP problem. To prevent cycling, one or more constraints are added to cut off the previously-found integer solution of the master problem.
Steps 3-5 are repeated until one of the termination criteria is satisfied.
A more detailed description of the general Outer Approximation algorithm can be found in [DG86].
Convexity and convergence
As linearizations are added to the master MIP problem, the model becomes an improved approximation of the original MINLP model. Using the usual convexity assumption regarding the nonlinear subproblem, convergence to a global optimum occurs when the objective function value of the master MIP problem is worse than the value associated with the NLP subproblem.
Several termination criteria are used in practice. These criteria can be used in isolation or in some logical combination. Three of them are discussed in the following paragraphs.
\(\ldots\) iteration limit
Perhaps the most frequently-used criterion is the iteration limit. One reason is that a good solution is usually found during the first few iterations. Another reason for using an iteration limit is that the size of the underlying master MIP problem tends to grow significantly each time linearization constraints are added, causing an increase in computation time.
\(\ldots\) objective worsening
A second criterion is the worsening of the objective function value of two successive nonlinear subproblems. This worsening occurs quite frequently, even if the NLP subproblem is convex. The underlying reason is that the master MIP problem will not always select binary solutions that lead to successively improving NLPs. This criterion seems appropriate when the worsening is persistent over several iterations.
A third termination criterion is insufficient improvement in the objective function value of the master MIP problem in relation to the objective function value of the previously solved NLP subproblem. The difference between these two values represents the optimality gap, since the master MIP problem represents an outer approximation (thus a relaxation) of the original MINLP model. When the gap is closed at crossover, the optimal solution has been found provided the NLP subproblem is convex.
Upon termination of the algorithm, the known best solution (also referred to as the incumbent solution) is declared as the final solution. In many practical applications, this solution is not necessarily optimal due to termination based on an iteration limit. In addition, it is often not possible to verify that the NLP subproblem is convex.
The term ‘outer approximation’ refers to the linear approximation of the convex nonlinear constraints at selected points along the boundary of the convex solution region. The accumulation of such inequality constraints forms an outer approximation of the solution region, and this approximation can be used in the optimization rather than the nonlinear constraints from which it was derived. The formula for the linearization of a scalar nonlinear inequality \(g(x,y) \leq 0\) around the point \((x,y) = (x^0,y^0)\) is as follows.
The nonconvex case
The linear approximation ceases to be an outer approximation if the solution region is not convex. In this situation there is the possibility that portions of the solution region are cut off as illustrated in Effect of loosening a linearization.
In practical implementations of the outer approximation algorithm, the linearizations are allowed to move away from the feasible region. Such heuristic flexibility allows solutions to be found that would otherwise have been cut off. The implementation allows deviations through the use of artificial nonnegative variables and then penalizing them while solving the master problem.
Open solver approach
The basic outer approximation algorithm that is part of the AOA module has been completely implemented using functionality provided by the GMP library.
From the math program instance representing the original MINLP model, a new math program instance representing the initial master MIP problem can be created using the function
The functions from the
GMP::Linearizationnamespace can be used to add linearizations of the nonlinear constraints of the original MINLP model to the master MIP, in a customizable manner.
GMP::Instance::FixColumnsprocedure, the integer columns of the nonlinear subproblem can fixed to the current integer solution of the master MIP.
GMP::Instance::AddIntegerEliminationRowsprocedure, prior integer solutions of the master MIP are excluded from subsequent solves.