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How to divide a territory? A new simple differential formalism for optimization of set functions
Author(s) -
Nguyen Hung T.,
Kreinovich Vladik
Publication year - 1999
Publication title -
international journal of intelligent systems
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.291
H-Index - 87
eISSN - 1098-111X
pISSN - 0884-8173
DOI - 10.1002/(sici)1098-111x(199903)14:3<223::aid-int1>3.0.co;2-7
Subject(s) - formalism (music) , simple (philosophy) , set function , computer science , set (abstract data type) , mathematical optimization , function (biology) , optimization problem , mathematics , algorithm , art , musical , philosophy , epistemology , visual arts , programming language , evolutionary biology , biology
In many practical problems, we must optimize a set function , i.e., find a set A for which f ( A )→max, where f is a function defined on the class of sets. Such problems appear in design, in image processing, in game theory, etc. Most optimization problems can be solved (or at least simplified) by using the fact that small deviations from an optimal solution can only decrease the value of the objective function; as a result, some derivative must be equal to 0. This approach has been successfully used, e.g., for set functions in which the desired set A is a shape , i.e., a smooth (or piece‐wise smooth) surface. In some real‐life problems, in particular, in the territorial division problem, the existing methods are not directly applicable. For such problems, we design a new simple differential formalism for optimizing set functions. ©1999 John Wiley & Sons, Inc.