Optimization of a function with only a few partial derivatives available

Question

Suppose you want to minimize a multivariate function where partial derivatives are available only wrt some of the variables but not all. Are there numerical optimizers designed for this case? Two obvious solutions would be:

1. Ignore the availability of some derivatives and use a derivative free optimizer (for example BOBYQA)
2. Numerically approximate the "missing" derivatives and use a gradient based optimizer

Has something better been tried?