Spline-approximation as the basis of computer technology design of linear structures routes

This article is a continuation of the article published in Journal of Applied Informatics nо.1 in 2019 [1]. In it, the problems of computer design of routes of various linear structures (new and reconstructed railways and highways, pipelines for various purposes, canals, etc.) are considered from a unified standpoint, as problems of approximating a sequence of points on plane of a smooth curve consisting of elements of a given type, i.e. spline. The fundamental difference from other approximation problems considered in the theory of splines and its applications is that the boundaries of the elements of the spline and even their number are unknown. Therefore, a two-stage scheme for finding a solution has been proposed. At the first stage, the number of spline elements and their parameters are determined using dynamic programming. For some tasks, this stage is the only one. In more complex cases, the result of the first stage is used as an initial approximation to optimize the spline parameters using nonlinear programming. Another complicating factor is the presence of numerous restrictions on the spline parameters, which take into account design standards and conditions for the construction and subsequent operation of the structure. The article discusses the features of mathematical models of the corresponding design problems. For a spline consisting of arcs of circles, mated by line segments, used in the design of the longitudinal profile of both new and reconstructed railways and highways and pipelines, a mathematical model is built and a new algorithm for solving a nonlinear programming problem is proposed, taking into account the structural features of the constraint system. In contrast to standard nonlinear programming algorithms, a basis is constructed in the zero-space of the matrix of active constraints and its modification is used when the set of active constraints changes. At the same time, to find the direction of descent at each iteration, no solution of auxiliary systems of equations is required at all. Two options for organizing the iterative optimization process are considered: descent through groups of variables in the presence of sections for independent construction of the descent direction and the traditional change of all variables in one iteration. Experimentally, no significant advantage of one of these options has been revealed.

OPTIMIZATION OF SPLINE PARAMETERS WHEN DESIGNING ROUTES OF LINEAR STRUCTURES

It considers the problem of approximating a discrete sequence of points on a plane by a spline consisting of line segments conjugated by circular arcs with unknown boundaries and the number of spline elements. This article is a continuation of the article published in N 6, 2019. This problem arises when designing the longitudinal profile of new and reconstructed railways and highways. The fundamental differences of the considered problem from the problems solved in the theory of splines and its applications are shown. A two-stage scheme is proposed: at the first stage, using a special dynamic programming algorithm, the number of elements of the spline and the approximate values of its parameters that satisfy all the constraints are determined. At the second stage, this result is used as an initial approximation to optimize the spline parameters using a special nonlinear algorithm. Significant simplifications of the algorithm of the first stage are implemented in comparison with the previously published one, due to the absence of clothoids when conjugating straight lines and curves. The necessity of the second stage in the design of new roads is substantiated to take into account the interconnection of spline elements in embankments and in excavations, if embankments will be constructed from excavation soil, and optimization is performed according to the criterion of minimum construction costs. A new nonlinear programming algorithm is proposed based on the construction of a basis in zero spaces of matrices of active constraints and the correction of this basis in an iterative process when changing the set of active constraints. It is shown how to find the direction of descent and solve the problem of excluding constraints from the active set without solving systems of linear equations in general or by solving linear systems of low dimension. Instead of the traditionally used sum of squares of deviations of the approximated points from the spline, other models are proposed as a model of the objective function, taking into account the specifics of a specific design problem.

Opportunities for Making Production-Related Decisions on the basis of Shadow Prices

This article describes how shadow prices can be used as active constraints (in this case constraints of mine production capacity)to address and support production-related decision-making. This is an algorithm from a post-optimal analysis developed by theauthor as part of a method for rationalising production decisions for a formal group (PGG, a company) of hard coal mines.Opportunities for using shadow prices are presented using examples of actual mines. The developed algorithm provides a quickway of obtaining information, with no need to solve the problem again, about possible gains or losses resulting from an increase ora decrease in a selected production limit, to determine how changes to such constraints will affect the profits and production andsales structures for specific coal sizes.

Existence of solution of constrained interval optimization problems with regularity concept.

Objective of this article is to study the conditions for the existence of efficient solution of interval optimization problem with inequality constraints. Here the active constraints are considered in inclusion form. The regularity condition for the existence of the Karush -Kuhn-Tucker point is derived. This condition depends on the interval-valued gradient function of active constraints. These are new concepts in the literature of interval optimization. gH -differentiability is used for the theoretical developments. gH -pseudo convexity for interval valued constrained optimization problems is introduced to study the sufficient conditions. Theoretical developments are verified through numerical examples.