Provably Convergent Multifidelity Optimization Algorithm Not Requiring High-Fidelity Derivatives

This paper presents a provably convergent multifidelity optimization algorithm for unconstrained problems that does not require high-fidelity gradients. The method uses a radial basis function interpolation to capture the error between a high-fidelity function and a low-fidelity function. The error interpolation is added to the low-fidelity function to create a surrogate model of the high-fidelity function in the neighborhood of a trust region. When appropriately distributed spatial calibration points are used, the low-fidelity function and radial basis function interpolation generate a fully linear model. This condition is sufficient to prove convergence in a trust region framework. In the case when there are multiple lower-fidelity models, the predictions of all calibrated lower-fidelity models can be combined with a maximum likelihood estimator constructed using kriging variance estimates from the radial basis function models. This procedure allows for flexibility in sampling lower-fidelity functions, does not alter the convergence proof of the optimization algorithm, and is shown to be robust to poor low-fidelity information. The algorithm is compared with a single-fidelity quasi-Newton algorithm and two first-order consistent multifidelity trust region algorithms. For simple functions the quasi-Newton algorithm uses slightly fewer high-fidelity function evaluations; however, for more complex supersonic airfoil design problems it uses significantly more. In all cases tested, our radial basis function calibration approach uses fewer high-fidelity function evaluations when compared with first-order consistent trust region schemes.