A Radial Basis Function-Based Optimization Algorithm with Regular Simplex Set Geometry in Ellipsoidal Trust-Regions

Abstract

In this paper, we investigate two ideas in the context of the interpolation-based optimization paradigm tailored to derivative-free black-box optimization problems. The proposed architecture maintains a radial basis function interpolation model of the actual objective that is managed according to a trust-region globalization scheme. We focus on two distinctive ideas. Firstly, we explore an original sampling strategy to adapt the interpolation set to the new trust region. A better-than-linear interpolation model is guaranteed by maintaining a well-poised supporting subset that pursues a near regular simplex geometry of $n+1$ points plus the trust-region center. This strategy improves the geometric distribution of the interpolation points whilst also optimally exploiting the existing interpolation set. On account of the associated minimal interpolation set size, the better-than-linear interpolation model will exhibit curvature, which is a necessary condition for the second idea. Therefore, we explore the generalization of the classic spherical to an ellipsoidal trust-region geometry by matching the contour ellipses with the inverse of the local problem hessian. This strategy is enabled by the certainty of a curved interpolation model and is introduced to accounts for the local output anisotropy of the objective function when generating new interpolation points. Instead of adapting the sampling strategy to an ellipsoid, we carry out the sampling in an affine transformed space. The combination of both methods is validated on a set of multivariate benchmark problems and compared with ORBIT.