Maryna Sazonova: Mathematical theory of continuous nonlinear optimal set partition (OPS) problems with arrangement of subset centers

Abstract:
Mathematical theory of continuous nonlinear problems of optimal partition of a set Ω in an n-measurable Euclidean space into disjoint subsets with arrangement of their centers is proposed.

Mathematical formulations of continuous nonlinear OPS problems with the arrangement of the centers of subsets under equality and inequality constraints or without constraints are formulated for cases of convex or concave nonlinear parts of the objective functional. The solution methods are substantiated for these problems. They are based on passing from the original nonlinear infinite-dimensional optimization problem through the Lagrangian functional and with application of the Kuhn–Tucker theory to the dual finite-dimensional problem with nonsmooth objective functional, and with the simultaneous obtaining of the analytical relations of direct and dual variables while solving the auxiliary operator equation with parameters. For solving the obtained nonsmooth dual task an objective functional is minimized by the method of generalized pseudogradients with space dilatation close to the Shor r-algorithm or its modifications.

The possibility of transferring and applying the theory of continuous nonlinear OPS problems with arrangement of subset centers to the case of corresponding continuous nonlinear OPS problems with fixed subset centers is shown.

Models of some applied infinite-dimensional problems of the arrangement of enterprises with simultaneous partition of a given region, continuously filled with customers, into domains of customers are developed. Each domain is serviced by one enterprise in order to minimize transportation and industrial costs. The customers may be telephone/ Internet subscribers, students, voters, points of an irrigated territory, patients to be diagnosed, etc. The NOPS system (which is a software implementation of all the above algorithms) was created and used for solving such problems both with the arrangement of the centers of subsets and with fixed centers of subsets. A comparative analysis of the results has shown that the methods that provide the possibility of center arrangement are more effective in optimizing the partitioning quality criterion.