### Handling Constraints

Practical optimization problems usually involve constraints, which can make the problems harder to solve. Constraints are handled in genetic algorithms in several ways.

#### Encoding Strategy

The simplest approach is to set the problem encoding, genetic operators, and initialization such that the constraints are automatically satisfied. Fixed constant bounds are easily handled in this manner in the GA procedure with the SetBounds call. The default initialization process and genetic operators provided by the GA procedure automatically respect bounds specified in this manner. For some types of constraints, you might be able to create a direct mapping from a constrained solution domain to a second domain with simple constant bounds. You could then define your genetic operator to map the solution into the second domain, apply one of the standard genetic operators, and then map the result back to the original domain.

If the problem contains equality constraints, you should try to transform the problem to eliminate the equality constraints and reduce the number of variables. This strategy is opposite from what is usually done in linear programming, where inequality constraints are turned into equality constraints by the addition of slack variables.

#### Repair Strategy

If the constraints are more complex and cannot be easily satisfied automatically by the genetic operators, you might be able to employ a repair strategy: check the solution and modify it to satisfy the constraints. The check and repair can be done in a user genetic operator when the solution is generated, or it can be done in the evaluation phase in a user objective function. Possible strategies for making a repair inside an objective function include projecting the solution onto the constraint boundary; while inside a genetic operator you might try adjusting an operator parameter until the constraint is satisfied. If you do the repair in the objective function, you should compute the objective value after performing the repair. You can write the repaired solution back out to the population with a WriteMember call from a user objective function or mutation subroutine, and with a WriteChild call from within a crossover subroutine. Example 3.2 illustrates the use of the repair strategy.

#### Penalty Strategy

Another technique is to permit solutions to violate constraints, but also to impose a fitness penalty that causes the population to evolve toward satisfying constraints as well as optimizing the objective. One way of employing this strategy is to simply add a penalty term to the objective function, but this approach should be used with care, because it is not always obvious how to construct the penalty function in a way that does not bias the optimization of the desired objective.

#### Direct Comparison Strategy

Using tournament selection opens another possibility for handling constraints. Define a fitness comparison routine (designated in a SetCompareRoutine call) that employs the following logic:

• If neither solution is feasible, choose the one closest to satisfying the constraints.

• If one solution is feasible, and the other is not, choose the feasible one.

• If both solutions are feasible, choose the one with the best objective value.

This strategy has the advantage that the objective function does not have to be calculated for infeasible solutions. To implement this method, you need to provide a measure of constraint violation and compute it in a user objective function; this value can be used in the first comparison step outlined previously. For linear constraints, the GA procedure provides the EvaluateLC call for this purpose. The technique works best when the solution space normally contains a significant number of solutions that satisfy the constraints. Otherwise it is possible that a single feasible solution might quickly dominate the population. In such cases, a better approach might be the following Bicriteria Comparison Strategy.

#### Bicriteria Comparison Strategy

A variation of the direct comparison strategy that has proved effective in many applications is the multiobjective, bicriteria approach. This strategy involves adding a second objective function, which is the magnitude of the constraint violation. Based on the original and constraint violation objective functions, a Pareto-optimal set of solutions is evolved in the population, and the Pareto-optimal set is evolved toward zero constraint violation. This technique is illustrated in Example 3.3. See the section Optimizing Multiple Objectives for a full discussion of Pareto optimality and how to apply this technique.