Minor Description
The following statement produces a NullPointerException:
new org.apache.commons.math3.optim.nonlinear.vector.jacobian.LevenbergMarquardtOptimizer().getWeight();
The documentation does not seem to indicate that other data must be set before getWeight is used (at least I could not find that information). In this case, weightMatrix is still null because it has not been initialized.
This call should probably throw an IllegalStateException, which makes it clear that this API usage is incorrect.
This test uses LevenbergMarquardtOptimizer but any instantiable subclass of MultivariateVectorOptimizer probably works the same way.