COMPUTER DESIGN OF ROUTES OF LINEAR STRUCTURES USING ALGORITHMS SPLINE-APPROXIMATIONS OF MULTIPLE-VALUED FUNCTIONS. Part 2

Abstract

The article is the end of the previously published article, which considered the first stage of solving the problem of approximation by a spline, consisting of arcs of circles conjugated by straight lines, a multivalued function given by a sequence of points on the plane. At this stage, using dynamic programming, the number of spline elements and the approximate values of its parameters are determined. This article considers the second stage: optimization of spline parameters using nonlinear programming algorithms. The result of the first stage is used as the initial approximation for the iterative process. Target function: the sum of the squared deviations of the given points from the desired spline. The variables that determine the position of the spline and the deviations of the given points from it are the lengths of line segments, circular arcs and their radii. These variables are subject to constraints in the form of inequalities. There may also be restrictions on deviations of individual points from the spline. An analysis of the features of the problem is given, the main of which is that the objective function is not expressed analytically in terms of variable lengths and radii. Nevertheless, formulas are given for calculating its partial derivatives with respect to these variables. A non-linear programming algorithm has been implemented, in which restrictions on element lengths and radii are taken into account when calculating the direction of descent, and restrictions (equalities and inequalities) for deviations of given points from the spline are taken into account using the modified Lagrange function (MFL). Due to the “ravine” nature of the MFL, the conjugate gradient algorithm is first used (for descending into the “ravine”), and then the wellknown DFP algorithm that uses the descent directions built on the iterations passed to construct a matrix approximate to the inverse matrix of second derivatives. The purpose of the article is to present a mathematical model and algorithms for optimizing the parameters of a spline approximating multivalued functions that represent the routes of linear structures. With regard to the design of structures such as trenches for laying pipelines for various purposes, irrigation network channels, the resulting spline is the final solution. With regard to the design of the railway. and a.d. the formulas for calculating partial derivatives given in the article have to be generalized to the case of a spline, in which circles are conjugated with straight lines using a clothoid.