Research On Innovative Optimization Algorithm Based On Polynomial Interpolation Approximation

Essentially,the solution of optimization problem is related to the global property of the problem.The analytic approximation using derivative information is a typically local asymptotic approximation.If the interpolation nodes are constructed appropriately,the interpolation approximation model can effectively overcome the above contradictions.The paper studies the optimization algorithm design based on polynomial interpolation approximation.The interpolation nodes are updated according to certain rules in an appropriately large neighborhood of the iteration points,then select the appropriate polynomial approximation models and give the corresponding iterative algorithms.These models will contribute to flexibility of approach,design space and research value.In the first two chapters,the theoretical background and typical architecture of optimization methods are summarized.Several important types of direct methods are particularly introduced.In the third chapter,using the idea of lightweight method in deep learning,we research the construction methods of various quadratic interpolation polynomial approximation models.The different algorithm designs presented in this paper mainly depend on the selection of these different approximation models.In the paper,the shapes of coefficient matrix of the approximation model are mainly considered,which are band,"closed" band,arrow,"closed"arrow and Hessenberg type.As a comparison,two typical interpolation algorithms,quadratic polynomial without cross term and complete quadratic polynomial,are also considered in the numerical test.Numerical experiments on a class of test functions verify the feasibility of these algorithms,and compare the advantages and disadvantages of each algorithm.Think about whether it’s a band matrix or an arrow matrix,essentially corresponds to a particular combination of different "pairs of variables" of all the variables.In chapter 4,a random strategy is introduced.At different iterations,the selected combination of intersecting variables interpolation points are determined by a randomly generated full arrangement of order N.Results of numerical test show that this improved strategy is more efficient than the algorithm given in Chapter 3.