Optimization of expensive black-box problems via Gradient-enhanced Kriging

Abstract

This paper explores the use of Gradient-enhanced Kriging for optimization of expensive black-box design problems, which is not completely limited by the conventional Efficient Global Optimization algorithm framework. Specifically, we give the best linear unbiased predictor and mean squared prediction error of the partial derivatives of Gradient-enhanced Kriging and then propose a measure named “Approximate Probability of Stationary Point” to estimate the approximate probability of a candidate infill point be a stationary point of the underlying function. When it comes to the selection of infill point, we not only maximize the well-known Expected Improvement but also evaluate the Approximate Probability of Stationary Point as a “double-check” step. Then the infill decision is made according to the extent of consistency between these two quantities. Furthermore, to examine whether the optimization process will gain from sparing more costs for response evaluation, we investigate also the cases that the gradient evaluation step is conditionally skipped in some iterations. Three new infill criteria are proposed and experimented with three analytical test functions and an airfoil optimal shape design. Results show that the optimization performance can be improved by exploiting the auxiliary gradient information in the proposed way.