The object of study is the process of mathematical modeling of combinatorial configurations. The purpose of the work is mathematical modeling of combinatorial configurations mapped into Euclidean space and the study of extreme problems on Euclidean combinatorial configurations. Research methods are: methods of mathematical modeling, set theory, algorithmic information theory, functional and combinatorial analysis, general algebra, Euclidean combinatorial optimization, nonlinear programming including convex and discrete programming, polyhedral and algebraic combinatorics, mathematical logic; Euclidean, affine and algebraic geometry. Theoretical and practical results are: in the dissertation, mathematical models and methods are developed and improved allowed to solve an important scientific problem of developing a general methodology for studying extreme problems on combinatorial configuration sets mapped into Euclidean space. Results of the work develop the Euclidean combinatorial optimization theory in its main directions: an expansion of a class of Euclidean combinatorial sets and a study of properties of their images in Euclidean space; investigating extremal properties of functions defined on them; developing new optimization methods; mathematical modeling of real-world problems as Euclidean combinatorial optimization problems. The scientific novelty of the results is: for the first time, a class of Euclidean combinatorial sets is sindgled out, which is a special class of sets of configurations in sense of Berge induced by vectors of the same dimension, which allow constructing mathematical models of optimization problems equivalent to a wide class of real-world problems; for the first time, new mathematical objects, called a Euclidean combinatorial configuration (e configuration) and a set of e configurations (C-set), are introduced; for the first time, a structural and geometric classification of C-sets is performed, which comprehensively uses in their modeling constructive features of their formation, a mappings specifics, and Euclidean space properties; for the first time, a class of basic C-sets (Cb-sets) is singled out, whose combinatorial structure can be expressed analytically by structural analysis means; it is used to systematize available information on the theory of combinatorial configurations and e-sets and extended these results; a number of new classes of Cb-sets are singled out and their algebraic topological and topological metric properties are investigated; for the first time, the main principles of the theory of minima estimates of functions defined on e-sets’ images in Euclidean space are systematized, extended and adapted to C-sets; the behavior of linear, quadratic, and convex functions on various classes of Cb-sets are investigated and their properties are derived; for the first time, a unified approach to analytical description of C-sets is offered as a way of their mathematical modeling by constructing their continuous functional representations (f-representations); a typology, construction and transformations methods of f-representations are introduced; f-representations of finite point configurations are found as mathematical models of some Cb-sets; the theory of convex extensions is developed in such directions as forming a general methodology for the formation of convex extensions from polyhedral-spherical sets, adaptation of the convex extensions theory to C-sets as extension domains, developing new approaches to forming convex extensions of functions from images of e-sets, extension of classes of extending functions, developing the general methodology for constructing functions’ extensions based on applying f-representations of C-sets as extension domains; combining modeling C-sets with the forming f-representations of the corresponding Cb-sets and extensions of functions from them; for the first time, a concept of constructing equivalent mathematical models of extreme combinatorial problems, which proves the possibility of using convex programming in theie solving them, is presented and theoretically substantiated; for the first time, a number of Euclidean statements of model problems are formed as optimization problems on C-sets including geometric design problems; the derived properties of extreme problems on C sets are used in developing existing tools and creating of new tools of Euclidean combinatorial optimization. The results of the dissertation work were implemented at the National Aerospace University named after N. E. Zhukovsky "Kharkiv Aviation Institute", Poltava National Technical Yuri Kondratyuk University, and Kharkiv National University of Radio Electronics. These results can be used in practical areas such as transport and warehousing logistics, info-communications, geometric design; in theoretical areas such as optimization theory, discrete mathematics, graph theory, polyhedral combinatorics, computational intelligence, etc.
