Radial Basis Function Response Surfacemethod Based On Product Auxiliary Function

The black-box problem refers to the problem that there is no clear expression between the input and the output,and it usually takes a lot of time or expensive to get the output.Such problems can only be solved by optimizing the information of function value,and they all aim to obtain the approximate solution of the global optimum of the original objective function by using as few function value evaluation times as possible.In recent years,optimization methods based on response surface models have been widely studied and applied in solving such problems.This dissertation focuses on the radial basis response surface method based on the product auxiliary function to solve black-box optimization problems.The main contents of this article are arranged as follows:Chapter 1 first introduces the black-box optimization problem and its research significance,and provides a review of the current state of research on several classes of derivative-free global optimization methods.Chapter 2 introduces the basic knowledge required to understand this dissertation,including algorithmic steps for response surface models,radial basis function interpolation models and an initial point design method,and finally presents the primary work of this dissertation.Chapter 3 addresses the problem of expensive black-box global optimization and proposes a response surface method based on product-assisted functions that can be adaptively sampled in an iteration.In terms of sampling point selection,a strategy of selecting the next sampling point based on the distance between the global optima of the response surface model in two consecutive iterations is combined with a strategy of using the global optima of the corresponding product auxiliary optimization problem as the next sampling point to obtain an adaptive sampling response surface method in order to better balance the global and local search.The algorithm is compared with several effective response surface methods in numerical experiments to obtain more satisfactory results.Chapter 4 adopts the convex combination of two radial basis functions as the response surface model under the framework of adaptive sampling response surface method.In order to further improve the fitting capability of the response surface model,and because both the cubic radial basis function and the thin plate sampling radial basis function have the characteristics of simple form,no hyperparameters and excellent fitting to the original function,a convex combination of these two radial basis functions is used as the response surface model.Numerical results on test problems verify the advantages of the algorithm.Chapter 5 first introduces the black-box optimization problem with expensive constraints and two algorithms for solving such problems.An algorithm for solving black-box optimization problems with expensive constraints is then proposed,which builds on the two-stage algorithm by applying the adaptive sampling strategy and the response surface combination strategy to the second stage of the algorithm.Numerical experiments demonstrate the effectiveness of the proposed algorithm in solving such problems.