Imposing Constraints on Design Variables

Quite often, design optimization problems involve semi-independent design variables. That is, some of the design variables may have to satisfy a certain relationship, but they vary independently. This would be true, for example, if you had three variables that were independent, but you wanted their sum to equal a certain value. In general, there are two ways to deal with these types of problems:

1. You can impose a constraint on the design variable values using a formula-based response.
2. You can redefine the design variables such that only designs that meet the imposed constraints can be created.

While the first approach is the simplest to set up, it will generate many infeasible designs, which will make the design space more difficult to search. Also, if the constraint is not evaluated until the analysis is complete, a significant amount of CPU time will be wasted analyzing infeasible designs.

The second approach involves some planning to establish formulas for the ranges. It may also require additional project variables. But the number of independent design variables will be same or possibly even reduced, depending on the form of the constraint. Furthermore, this representation yields a better posed optimization problem, so you are likely get better results.

For example, imagine you are attempting to calibrate a material model that has three parameters –  C1, C2, and C3 – and that the material model is only stable when: C1 ≥ C2 ≥ C3 ≥ 0. Using the second approach (redefining the variables), the variable definition might look like this:

1. Define C1 as a continuous variable with whatever range makes sense. In this case, assume C1 can take on any value in the range 0 to 5.
2. Define two new variables, p2 and p3, that can take on any value in the range 0 to 1.
3. Define C2 as a dependent variable using the formula C2 = p2*C1.
4. Define C3 as a dependent variable using the formula C3 = p3*C2.

This definition will ensure that the relationship between the design variables is always true for any value of the independent variables C1, p2 and p3 within their respective ranges. Note that the number of actual design variables used in the search did not change, though some other types of relationships could result in a reduced number of design variables.

In summary, when design variables are semi-independent, we recommend that you redefine the problem in a manner that will yield only solutions that meet the desired relationship(s) among the variables. This is the preferred representation of the problem. This approach may require a slightly more complex optimization setup, but it makes the search much more productive and increases the chance of finding better solutions.

More examples of how to use this technique are provided here.