Наближений метод пошуку екстремуму функціонала на множині перестановок

Abstract

Розглянуто окремий клас дискретних задач геометричного проектування. Наведено формальну постановку задачі призначення геометричних об’єктів у вигляді задачі оптимізації на множинах перестановок. Запропоновано один із наближених методів пошуку екстремуму функціонала на множині перестановок.

The high computational complexity of the combinatorial optimization methods, the difference of the combinatorial properties of the sets which form the ranges of admissible solutions, are the reasons for the lack of unified approach to combinatorial optimization problems solving. The basic idea of the combinatorial methods consists in the transition from complete enumeration of finite set of solutions to reduced one. The impossibility of exact solution of combinatorial optimization problems of large dimension and specific limitations cause the development of approximate methods, but these methods also have serious disadvantages such as the obtained local extremum may not coincide with the global one, it is impossible to estimate the difference between the local and global extremum a priori. On this base, the development of optimization methods for various classes of functions on combinatorial sets is the topical problem. The unified approach to the study of geometric design problems on the base of the formalization of the concept of geometric information and the introduced information space is proposed in the research. In the research the main attention is given to the problem of locating geometric objects, constructing of the mathematical model of this problem. The solution of the discrete geometric design problem is proposed with use of the method which bases on immersing of combinatorial sets in arithmetic Euclidean space. The formulation of the practical problem of geometric design is presented.