Nonlinear and Dynamic Programming Methods for Solving the Variational Problems of a Special Type

Abstract

In linear structures routing we have the following problem : find the extremal of given functional, i.e 2D or 3D curve , which must consist of a special type elements. The parameters of elements are limited and their number is unknown. At first we must determine number of elements and after this we can find their optimal parameters. In the case of 2D extremal we shall consider a broken line and parabolic spline. The broken line is used as a longitudinal profile of railways and the parabolic spline is used as a longitudinal profile of roads. Initial problem was solved as multi-stage process using the methods of dynamic and nonlinear programming. On each stage we consider the different mathematical models of unknown extremal line: 1. A broken line with short elements similar to longitudinal profile of ground. This model allow us to find the initial approximation of unknown line using nonlinear programming. 2. The result of first stage give us opportunity to find number of elements using dynamic programming. 3. We use a special algorithm of nonlinear programming for solving initial problem with fixed number of elements and the result of second stage as initial approximation.