A derivative-free trust-region algorithm for unconstrained optimization with controlled error

Multi-fidelity algorithms for the horizontal alignment problem in road design

“…As the algorithm converges to better points in the search space, the surrogate model utilizes the higher accuracy. Examples of this technique can be found in [28,31,36].…”

“…They showed that it can solve the problem more quickly than using the algorithm with only the high fidelity model. Another example of multi-fidelity algorithms used in other fields is the implied volatility estimation where authors [31] showed that using a trust region algorithm based on the support vector regression model, they can solve the problem much faster than surrogate-free version. Using the idea of multi-fidelity, they proved that the algorithm converges to the stationary point.…”

“…In this thesis, we apply both GPS with adaptive control [28] and the trust region algorithm with controlled error [31] to solve the horizontal problem in road design. The algorithms we choose fall into multi-fidelity hierarchical methods as the algorithms do not employ high fidelity models at each iteration, and they use the high fidelity models when they are close enough to the stationary point.…”

“…It can be created randomly or using a positive basis [4]. For example, Takaki [31] used a random method to generate sample points. In [28], authors suggested constructing sample points using the canonical maximal positive basis (Definition 2.7).…”

“…If the number of remaining points in X is less than N l , new points are generated until the number of points in X is more than or equal to N l . The new points can be generated using random sampling [31], Latin Hypercube sampling [19] or using positive basis [28].…”