Research On Optimization Algorithms Based On Polynomial Interpolation Approximation

This article focuses on the design of optimization algorithms for a special type of polynomial approximation model.The essence of polynomial interpolation approximation model is to construct an approximate model of the objective function through interpolation points,which is a direct method that does not use derivative information of the objective function.Direct methods give up the limitation of the local asymptotic characteristics based on derivative information and can reflect better global optimization performance through the characteristics of interpolation nodes.In some cases,direct methods have advantages over analytical methods.This article first proposes an improved three-point onedimensional search algorithm.The improved algorithm proposes a new way of selecting trial points,and the one-dimensional search algorithm obtained by this method converges faster than classical interval partitioning algorithms.The article also defines the central variable of the interpolation polynomial model,proposes three methods for constructing central variables,and designs two different structures according to the different quadratic coefficient matrices of the interpolation model.Based on the central variables and the structure of the interpolation model,suitable interpolation point and interpolation radius construction methods are designed,and specific algorithm implementations are given.The improved interpolation polynomial model achieves a good balance between the utilization of objective function information and computational efficiency.Numerical experiments are conducted to compare the proposed algorithm with classical interpolation models.It is found that the improved interpolation model has advantages such as fewer iteration steps and higher solution accuracy when facing high-dimensional objective functions.