Application of the Theory of Optimal Set Partitioning for Constructing Fuzzy Voronoi Diagrams

Abstract

AbstractThe paper demonstrates the possibility of constructing fuzzy Voronoi diagrams based on a unified approach. This approach implies formulating a continuous problem of optimal partitioning of a set from n-dimensional Euclidean space into subsets with a quality criterion that provides the appropriate form of the Voronoi diagram. The theory of optimal set partitioning is a universal mathematical apparatus for constructing Voronoi diagrams, which is based on the following general idea. The original problems of optimal set partitioning, which are mathematically formulated as infinite-dimensional optimization problems, are reduced through the Lagrange functional to auxiliary finite-dimensional nonsmooth maximization problems or to nonsmooth maximin problems, for the numerical solution of which modern efficient optimization methods are used. A feature of this approach is the fact that the solution of the original infinite-dimensional optimization problems can be obtained analytically in an explicit form, and the analytical expression can include parameters sought as the optimal solution of the above auxiliary finite-dimensional optimization problems with nonsmooth objective functions. The paper proposes an algorithm for constructing one of the variants of fuzzy Voronoi diagrams, when the set of points forming a Voronoi cell may be fuzzy. The algorithm is developed based on the synthesis of methods from the theory of optimal set partitioning and the theory of fuzzy sets. The proposed algorithm is implemented in software; its work is demonstrated by examples of constructing standard and additively weighted diagrams with fuzzy Voronoi cells.KeywordsFuzzy Voronoi diagramGenerating pointsOptimal set partitioningNondifferentiable optimizationShor’s r-algorithm