A radial basis function method for noisy global optimisation.

Area of Science:

• Numerical analysis
• Optimization
• Computational mathematics

Background:

• Global optimization of expensive and noisy black-box functions is challenging.
• Existing methods like Gutmann's Radial Basis Function (RBF) method are effective for deterministic functions but require adaptation for noisy scenarios.
• The availability of error bounds for noisy function evaluations is crucial for robust optimization.

Purpose of the Study:

• To develop a novel response surface method for the global optimization of expensive and noisy black-box objective functions.
• To incorporate available error bounds into the optimization process for improved accuracy and reliability.
• To extend the principles of Gutmann's RBF method to handle noisy function evaluations.

Main Methods:

• The proposed method utilizes a regularised least-squares criterion for constructing Radial Basis Function (RBF) approximants.
• It determines new sample points for successive evaluations based on a target value, analogous to the original RBF method.
• Error bounds on noisy function values are integrated into the approximation and sampling strategy.

Main Results:

• The method provides a robust framework for global optimization of expensive, noisy objective functions.
• Convergence properties of the novel response surface method are established.
• A numerical illustration on a test problem demonstrates the method's practical applicability.

Conclusions:

• The developed response surface method effectively handles expensive and noisy black-box optimization problems.
• The integration of error bounds enhances the reliability of the global optimization process.
• The method offers a valuable extension to existing RBF-based optimization techniques.