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International Conference on Parallel Computational Technologies

PCT 2017: Parallel Computational Technologies pp 86–100Cite as

On the Solution of Linear Programming Problems in the Age of Big Data

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Part of the book series: Communications in Computer and Information Science ((CCIS,volume 753))

Abstract

The Big Data phenomenon has spawned large-scale linear programming problems. In many cases, these problems are non-stationary. In this paper, we describe a new scalable algorithm called NSLP for solving high-dimensional, non-stationary linear programming problems on modern cluster computing systems. The algorithm consists of two phases: Quest and Targeting. The Quest phase calculates a solution of the system of inequalities defining the constraint system of the linear programming problem under the condition of dynamic changes in input data. To this end, the apparatus of Fejer mappings is used. The Targeting phase forms a special system of points having the shape of an n-dimensional axisymmetric cross. The cross moves in the n-dimensional space in such a way that the solution of the linear programming problem is located all the time in an \(\varepsilon \)-vicinity of the central point of the cross.

The reported study has been partially supported by the RFBR according to research project No. 17-07-00352-a, by the Government of the Russian Federation according to Act 211 (contract No. 02.A03.21.0011) and by the Ministry of Education and Science of the Russian Federation (government order 2.7905.2017/8.9).

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Notes

  1. 1.

    A single-valued map \(\varphi :{\mathbb {R}^n} \rightarrow {\mathbb {R}^n}\) is said to be fejerian relatively to a set M (or briefly, M-fejerian) if

    $$\begin{aligned} \begin{array}{l} \varphi \left( y \right) = y, \forall y \in M;\\ \left\| {\varphi (x) - y} \right\| < \left\| {x - y} \right\| , \forall x \notin M,\forall y \in M.\\ \end{array} \end{aligned}$$

    .

  2. 2.

    Here \(\mathrm {dist}(z,M) = \inf \left\{ {\left\| {z - x} \right\| :x \in M} \right\} \).

  3. 3.

    The symbol \( \div \) denotes integer division.

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Authors and Affiliations

  1. South Ural State University, 76 Lenin Prospekt, Chelyabinsk, 454080, Russia

    Irina Sokolinskaya & Leonid B. Sokolinsky

Authors
  1. Irina Sokolinskaya
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  2. Leonid B. Sokolinsky
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Correspondence to Leonid B. Sokolinsky .

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Editors and Affiliations

  1. South Ural State University, Chelyabinsk, Russia

    Leonid Sokolinsky

  2. South Ural State University, Chelyabinsk, Russia

    Mikhail Zymbler

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Sokolinskaya, I., Sokolinsky, L.B. (2017). On the Solution of Linear Programming Problems in the Age of Big Data. In: Sokolinsky, L., Zymbler, M. (eds) Parallel Computational Technologies. PCT 2017. Communications in Computer and Information Science, vol 753. Springer, Cham. https://doi.org/10.1007/978-3-319-67035-5_7

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  • DOI: https://doi.org/10.1007/978-3-319-67035-5_7

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