Fast and exact Hessian computation for a class of nonlinear functions used in radiation therapy treatment planning

In nonlinear optimisation, using exact Hessian computations (full-Newton) hold superior convergence properties over quasi-Newton methods or gradient-based methods. However, for medium-scale problems, computing the Hessian can be computationally expensive and thus time-consuming. For solvers dedicated to a specific problem type, it can be advantageous to hard-code optimised implementations to keep the computation time to a minimum. In this paper we derive a computationally efficient canonical form for a class of additively and multiplicatively separable functions. The major computational cost is reduced to a single multiplication of the data matrix with itself, allowing simple parallellisation on modern-day multi-core processors. We present the approach in the practical application of radiation therapy treatment planning, where this form appears for many common functions. In this case, the data matrices are the dose-influence matrices. The method is compared against automatic differentiation.