I've noticed that defining unneccesary partial derivatives can significantly slow down the optimizer. Therefore I'm trying to understand: how can I know whether I should define the partial derivative for a certain input/output relationship?
When you say "unnecessary" do you mean partial derivatives that are always zero?
Using declare_partials('*', '*'), when a component is really more sparse than that will significantly slow down your model. Anywhere where a partial derivatives is always zero, you should simply not declare it.
Furthermore, if you have a vectorized operation, then your Jacobian is actually a diagonal matrix. In that case, you should declare a [sparse partial derivative] by giving rows and cols arguments to the declare_partial call1. This will often substantially speed up your code.
Technically speaking, if you follows the data path from all of your design variables, through each components, to the objective and constraints, then any variable you passed would need to have its partials defined. But practically speaking you should declare and specify all the partials for every output w.r.t. every input (unless they are zero), so that changes to model connectivity don't break your derivatives.
It takes a little bit more time to declare your partials more sparsely, but the performance speed up is well worth it.