Abstract
Under study is the problem of searching the extremal of given functional ( two or three-dimensional curve ), which must satisfy a number of predetermined restrictions. A specific feature of the desired curve is that it should consist of elements of special types, whose parameters are limited. The number of elements is unknown and should be determined in the process of solving the problem. Such problems arise in particular in designing linear structures routes. In the case searching of two-dimensional extremal piecewise linear and piecewise parabolic curves are considered. Such problems arise in the design of optimal longitudinal profile of railways and roads. Multi-stage approach is proposed using the methods of nonlinear and dynamic programming. At the first stage we define broken line consisting of elements of small length, using nonlinear programming. At the second stage we determine a number of the elements and the initial approximation, using dynamic programming. At the third stage we find the optimal decision using a special non-linear programming algorithm.
Highlights
Many of optimization problems from different practice areas are reduced to finding two or three dimensional curves, which must satisfy the conditions of smoothness and other restrictions
We want to find the extremal of the given functional, satisfying the system of restrictions
Examples of such tasks can serve as the search of optimal routes proposed roads and other linear structures, both at the stage of new construction and reconstruction phase
Summary
Many of optimization problems from different practice areas are reduced to finding two or three dimensional curves, which must satisfy the conditions of smoothness and other restrictions. In particular in the conditions of rugged relief and complex geology the cost of construction and subsequent operation can be significantly reduced at the optimum location of the projected route of the road on the ground It was established in the operation of the first CAD, in which the design of the longitudinal profile of new railways was carried out with the use of optimization program [ 1 ]. The problem of developing adequate mathematical models and mathematically correct algorithms optimize routes of new roads remains relevant This is the main way of improving CAD of linear structures. The aim of the study was to develop a mathematical model, methods and algorithms for solving variational problems which have important features, to improve CAD linear structures by using computer for optimization of their routes
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