Differentiable evaluation of objective functions in sampling design with variance-covariance matrices

In this short note we consider the differentiable evaluation of the objective function of the sampling design optimization problem based on the inverse of the Fisher information matrix, and where the integer design variables have been converted into real variables using a relaxation technique. To ensure differentiability and cover the full range of the variables, and thus improve the convergence behavior of derivative-based optimization algorithms, we propose applying a Cholesky decomposition on the Fisher information matrix, but using a special higher precision floating point arithmetic to ensure stability. While each evalu-ation of the functional becomes slower, the algorithm is much simpler, amenable to be used with automatic differentiation directly, and can be shown to be very stable. For many practical situations, this is a valuable trade-off. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)