Optimization of spline parameters in approximation of multivalued functions

Abstract

Objectives. Methods for spline approximation of a sequence of points in a plane are increasingly used in various disciplines. A spline is defined as a single-valued function consisting of a known number of repeating elements, of which the most widely used are polynomials. When designing the routes of linear structures, it is necessary to consider a problem with an unknown number of elements. An algorithm implemented for solving this problem when designing a longitudinal profile was published earlier. Here, since the spline elements comprise circular arcs conjugated by line segments, the spline is a single-valued function. However, when designing a route plan, the spline is generally a multivalued function. Therefore, the previously developed algorithm is unsuitable for solving this problem, even if the same spline elements are used. The aim of this work is to generalize the obtained results to the case of approximation of multivalued functions while considering various features involved in designing the routes of linear structures. The first stage of this work consisted in determining the number of elements of the approximating spline using dynamic programming. In the present paper, the next stage of solving this problem is carried out.Methods. The spline parameters were optimized using a new mathematical model in the form of a modified Lagrange function and a special nonlinear programming algorithm. In this case, it is possible to analytically calculate the derivatives of the objective function with respect to the spline parameters in the absence of its analytical expression. Results. A mathematical model and algorithm were developed to optimize the parameters of a spline as a multivalued function consisting of circular arcs conjugated by line segments. The initial approximation is the spline obtained at the first stage.Conclusions. The previously proposed two-stage spline approximation scheme for an unknown number of spline elements is also suitable for approximating multivalued functions given by a sequence of points in a plane, in particular, for designing a plan of routes for linear structures.