A Radial Basis Function based Optimization Algorithm with Regular Simplex set geometry in Ellipsoidal Trust-Regions

TL;DR

This paper introduces a derivative-free optimization algorithm using a novel sampling strategy within trust-region methods, employing Universal Kriging for surrogate modeling and extending trust-region geometry from spherical to ellipsoidal to handle anisotropy.

Contribution

It develops an original sampling strategy ensuring well-poised subsets with regular simplex geometry and generalizes trust-region geometry to ellipsoids for better local adaptation.

Findings

Improved scattering of sample points compared to existing methods.

Guarantees surrogate model curvature through sampling strategy.

Validated performance on multidimensional benchmark problems.

Abstract

We present a novel derivative-free interpolation based optimization algorithm. A trust-region method is used where a surrogate model is realized via an interpolation framework. The framework for interpolation is provided by Universal Kriging. A first contribution focuses on the development of an original sampling strategy. A valid model is guaranteed by maintaining a well-poised subset that exhibits the regular simplex geometry approximately. It follows that this strategy improves the scattering of points with respect to the state-of-the-art and, even importantly, assures that the surrogate model exhibits curvature. A second contribution focuses on the generalization of the spherical trust-region geometry to an ellipsoidal geometry, that to account for local anisotropy of the objective function and to improve the interpolation conditions as seen from the output space. The ensemble method is validated against its direct competitors on a set of multidimensional problems.