# The scheme complexity of discrete optimization

The scheme complexity of discrete optimization -- We consider an approach to the problem of minimization of a linear form (a, x) over £ M C 5J, where a G Rn is an input vector, M = N/ = {: f ( x ) = 1), f ( x ) is a characteristic function of fc-valued logic. To estimate the complexity of search of the optimal point, we use the structure characteristic of the set /, which is equal to the complexity of the scheme description of this set by means of functional elements in suitable bases. The complexity of descriptive schemes describing N/ and such that they are associated by a natural way with optimization schemes of the same complexity is also used. It is shown that there are essential analogues with the problems of synthesis of computing schemes from functional elements including the applicability of analogues of methods of the theory of control system to discrete optimization. In many cases this approach gives polynomial algorithms, not always being the best. Sometimes we may obtain a polynomial algorithm by means of a preliminary processing of the input vector a. This article is an extended version of the author's lecture on DC All-Union Conference http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Discrete Mathematics and Applications de Gruyter

# The scheme complexity of discrete optimization

Discrete Mathematics and Applications, Volume 3 (2) – Jan 1, 1993
20 pages

/lp/de-gruyter/the-scheme-complexity-of-discrete-optimization-JYTB8FJdvO
Publisher
de Gruyter
ISSN
0924-9265
eISSN
1569-3929
DOI
10.1515/dma.1993.3.2.127
Publisher site
See Article on Publisher Site

### Abstract

-- We consider an approach to the problem of minimization of a linear form (a, x) over £ M C 5J, where a G Rn is an input vector, M = N/ = {: f ( x ) = 1), f ( x ) is a characteristic function of fc-valued logic. To estimate the complexity of search of the optimal point, we use the structure characteristic of the set /, which is equal to the complexity of the scheme description of this set by means of functional elements in suitable bases. The complexity of descriptive schemes describing N/ and such that they are associated by a natural way with optimization schemes of the same complexity is also used. It is shown that there are essential analogues with the problems of synthesis of computing schemes from functional elements including the applicability of analogues of methods of the theory of control system to discrete optimization. In many cases this approach gives polynomial algorithms, not always being the best. Sometimes we may obtain a polynomial algorithm by means of a preliminary processing of the input vector a. This article is an extended version of the author's lecture on DC All-Union Conference

### Journal

Discrete Mathematics and Applicationsde Gruyter

Published: Jan 1, 1993

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