Grebennik I. Mathematical models and methods of combinatorial optimization in geometric design.

A process of technical systems, objects and processes geometric design is investigated. The thesis is devoted to generalization and development of Euclidean combinatorial optimization theory, construction of mathematical models and development of solution methods for the combinatorial optimization problems of geometric design. New combinatorial sets being composition k-images of combinatorial sets are introduced. Means for description and a classification of such sets are offered. An approach to linear function optimization on the composition images of combinatorial sets is proposed. The extreme properties and estimations of minimum of convex functions on the classes of the composition images of combinatorial sets are investigated. A mathematical model of the basic geometric design problem is constructed in an interval form. A method for the combinatorial optimization problems with a linear objective function and linear constraints. The basic optimization problems on interval combinatorial sets classesare solved. A solution method of the interval combinatorial optimization problems with quasilinear constraints, which based on transformation of ones to the combinatorial optimization problems in the Euclidean space, is proposed. Outcome of the thesis was introduced into such State enterprises: "Ukrtelecom", "Ukrpochta", "Meridian", "Potential". The results may be also applied to solve of problems of packing, placement, design, and team planning and educational process in different branches.