  In this work we propose two Hermite-type optimization methods, Hermite least
squares and Hermite BOBYQA, specialized for the case that some partial
derivatives of the objective function are available and others are not. The
main objective is to reduce the number of objective function calls by
maintaining the convergence properties. Both methods are modifications of
Powell's derivative-free BOBYQA algorithm. But instead of (underdetermined)
interpolation for building the quadratic subproblem in each iteration, the
training data is enriched with first and -- if possible -- second order
derivatives and then (weighted) least squares regression is used. Proofs for
global convergence are discussed and numerical results are presented. Further,
the applicability is verified for a realistic test case in the context of yield
optimization. Numerical tests show that the Hermite least squares approach
outperforms classic BOBYQA if half or more partial derivatives are available.
In addition, the Hermite-type approaches achieve more robustness and thus
better performance in case of noisy objective functions.
