МЕТОДЫ ЧИСЛЕННОЙ МИНИМИЗАЦИИ ФУНКЦИОНАЛА КВАЗИПРОТЯЖЕННОСТИ

Abstract

The data processing problems formulation based of the minimum-extent criterion leads to the problems of minimizing the functional of quasi-extent. Due to the impossibility of using analytical methods of calculus of variations, the successful solutions of these problems depend on the effectiveness of numerical minimization methods. In this paper, the problem of developing methods for one-dimensional and multidimensional numerical minimization of the quasi-extent functional, which is built on the residual function between the data and their model, is considered. The goal of this work is to create the effective methods for numerically minimizing the quasi-extent functional.In accordance to the problem statement, the methods of one-dimensional and multidimensional numerical minimization are considered. The main requirements to the methods of one-dimensional minimization of the quasi-extent functional were consisted both in the strict fulfillment of the condition for non-increasing of objective function formed on the basis of the quasi-extent functional as well as in the fulfillment of the condition of searching the several local minima. For this reason, the proposed one-dimensional minimization methods were based on the zero-order optimization methods with a priori or a posteriori choice of the feasible region of the objective function and with a selection of the set of basic local minima. The results of numerical simulation showed that the proposed methods allow solving successfully the problem of determining the values of single unknown data model parameter which is included in the data model linearly or nonlinearly and accepted one or more values. The main requirements to the methods of multidimensional minimization of the quasi-extent functional, which is depended on several unknown linear parameters, were consisted both in the fulfilling of the condition for non-increasing of objective function formed on the basis of the quasi-extent functional, as well as in the fulfillment of the condition of searching the global minimum. Due to these requirements, the proposed multidimensional minimization method was based on the conjugate gradient method, in which the one-dimensional minimization problem along the descent direction is solved by using the set of “test” steps. The results of numerical simulation confirmed the performance of proposed method.