Nedobachij S. Models, methods and algorithms in Euclidean combinatorial optimization problems.

The convex hull covers domains of determination of problems, models of which are Euclidean optimization problems on permutable sets, are investigated. The objective of the work is to determine new properties of given numbers and its convex hull cjvers, elaboration of a new method of exact solving the problem of minimization of weighed length of a linking net in linear location of rectangular elements. It has been found the irreducible systems of linear restrictions for the general permutable and the general polypermutable polyhedrons. For each of them it has been determined the equation of hyperfacets expressed through nonabundant limitations, obtained their quantity, described vertexes by eguations of hyperfacets. The number of hyperfacets that join on the same vertex has been determined. The new proof of coincidence of vertexes of general permutable polyhedron with a number of permutations with repetitions, by which it is induced, is given. The dependence between the number of hyperfacets of polyhed ron sum and the number of polyhedron hyperfacets that make this sum has been found. For permutable polyhedron the least number of ribs that join its two arbitrary vertexes has been determined. The algorithm of finding the way between two arbitrary vertexes which consist of tne least number of ribs is given. Thr properties of one reflection of polynomial permutations of first n natural numbers into a number of permutations of modules of their differences that appears in a prroblem of minimization of a weighed length ofalinking net in a linear disposition of rectangular elements has been proved. The new method of exact solution of this problem is given. Theobtained results can be used in a theory of Euclidean combinatorial optimization with its following development and during working out new algorithms of solution of practical optimization problems.