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Supermodular programming on finite lattices.

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Title: Supermodular programming on finite lattices.
Authors: Khachaturov, Vladimir1 rv_khach@yahoo.ie, Khachaturov, Roman1, Khachaturov, Ruben1
Source: Computational Mathematics & Mathematical Physics. Jun2012, Vol. 52 Issue 6, p855-878. 24p.
Subjects: Modular programming, Lattice theory, Boolean functions, Vector spaces, Modular functions, Mathematical analysis
Abstract: There is a discription of the problems of minimization of supermodular functions on the different types of lattices: Boolean lattices, lattices with relative supplements (division lattices, lattices of vector subspaces of finite-dimensional vector space, geometrical lattices), lattices equal to Cartesian product of chains. The previously obtained theoretical results, on the basis of which the problems of minimization of supermodular functions on these lattices have been solved, are shown. A new type of lattices, lattice of Cubes, is defined and described. The problems of minimization and maximization of supermodular functions are considered on it. Particular examples of such functions are given. Optimization algorithms and the possibilities of setting and solving a new class of problems on the lattices of Cubes are discussed. [ABSTRACT FROM AUTHOR]
Copyright of Computational Mathematics & Mathematical Physics is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
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  Data: There is a discription of the problems of minimization of supermodular functions on the different types of lattices: Boolean lattices, lattices with relative supplements (division lattices, lattices of vector subspaces of finite-dimensional vector space, geometrical lattices), lattices equal to Cartesian product of chains. The previously obtained theoretical results, on the basis of which the problems of minimization of supermodular functions on these lattices have been solved, are shown. A new type of lattices, lattice of Cubes, is defined and described. The problems of minimization and maximization of supermodular functions are considered on it. Particular examples of such functions are given. Optimization algorithms and the possibilities of setting and solving a new class of problems on the lattices of Cubes are discussed. [ABSTRACT FROM AUTHOR]
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  Data: <i>Copyright of Computational Mathematics & Mathematical Physics is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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              Text: Jun2012
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