Funcrional and analytic representations of the total permutation
DOI:
https://doi.org/10.15587/1729-4061.2016.58550Keywords:
functional representation of a set, total permutation, permutation polyhedron, combinatorial optimizationAbstract
The study introduces the notion of a functional representation of a set of points in the Euclidean space, suggests a classification of such representations, as well as describes the polyhedral-surface and the surface-polyhedral approaches to the functional representations of the total permutation. The study proves the existence of both strict and non-strict functional representations of the set, suggests a number of strict representations that are based on the studied properties of the two classes of symmetric functions, and presents the visualization and analysis of strict representations of small permutation arrays.
The study uses total permutations to present a new continuous reformulation of nonlinear tasks, which allows devising exact and approximate optimization algorithms that have the following properties:(a) they are based on the method of penalty function and a modified Lagrange method, (b) they do not use restrictions on the permutation polyhedron, and (c) they can be combined with convex extensions of objective functions from the set to a CD.
The obtained strict representations form a framework for new continuous relaxations of the combinatorial set and its subsets. Consequently, they can be the basis for new optimization methods and algorithms of total permutations. The methods can be applied to numerous practical problems if they are formulated as nonlinear unconditional tasks on permutations and their generalizations.References
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