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Large-Scale Problems with Quasi-Block Matrices

  • COMPUTER METHODS
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Journal of Computer and Systems Sciences International Aims and scope

Abstract

Sparse large matrices with a block-staircase and block-treelike structure are studied. They are called quasi-block matrices and consist of independent blocks that are connected to each other pairwise or in a more general fashion. The interdependence of parameters of such matrices, such as the number of nonzero elements, the number of blocks, and the matrix size, is determined. Integer programming problems with large quasi-block matrices are described. For the efficient solution of these problems, a local elimination algorithm is used. This is an iterative algorithm in which certain variables are eliminated at each step. The issues concerning the optimal elimination order are studied. This problem turns out to be exponentially complex, which is proved using a graph interpretation of the concepts of block-treelike and block-staircase structures. The complexity of the local elimination algorithm is considered. This is important for deciding which methods are better in different situations. The numerical results are presented; in particular, efficient procedures for determining the optimal elimination order are described. Special attention is given to the parallelization of particular quasi-block Boolean programming problems on a computer GRID if these problems cannot be solved on a single processor due to their large size.

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REFERENCES

  1. V. I. Tsurkov, Decomposition in Large-Scale Problems (Nauka, Moscow, 1981) [in Russian].

    MATH  Google Scholar 

  2. I. L. Averbakh and V. I. Tsurkov, Optimization in Large Problems with Integer Variables (Nauka, Fizmatlit, Moscow, 1995) [in Russian].

  3. V. I. Tsurkov, “The principle of decomposition for block-separable systems,” Dokl. Akad. Nauk SSSR 246, 27–31 (1979).

    MathSciNet  Google Scholar 

  4. O. A. Shcherbina, D. V. Lemtyuzhnikova, and V. I. Tsurkov, Multidimensional Problems with Quasi-Block Matrices (Fizmatlit, Moscow, 2018) [in Russian].

    Google Scholar 

  5. Yu. Yu. Finkel’shtein, Approximate Methods and Applied Problems of Discrete Programming (Nauka, Moscow, 1976) [in Russian].

    Google Scholar 

  6. D. V. Kovkov and D. V. Lemtyuzhnikova, “Decomposition in multidimensional boolean-optimization problems with sparse matrices,” J. Comput. Syst. Sci. Int. 57, 97 (2018).

    Article  MathSciNet  MATH  Google Scholar 

  7. S. Vigerske and A. Gleixner, “SCIP: global optimization of mixed-integer nonlinear programs in a branch-and-cut framework,” Optim. Methods Software 33, 563–593 (2018).

    Article  MathSciNet  MATH  Google Scholar 

  8. D. V. Lemtyuzhnikova and D. V. Kovkov, “Problems of discrete optimization with quasi-matrix matrices,” Int. J. Open Inform. Technol., No. 10, 1–8 (2017).

  9. Yu. I. Zhuravlev, Selected Scientific Works (Magistr, Moscow, 1998) [in Russian].

    Google Scholar 

  10. O. A. Shcherbina, “Local elimination algorithms for sparse discrete optimization problems,” Doctoral (Phys. Math.) Dissertation (Comput. Center of RAS, Moscow, 2010).

  11. M. Yannakakis, “Computing the minimum fill-in is NP-complete,” SIAM J. Alg. Disc. Meth. 2, 77–79 (1981).

    Article  MathSciNet  MATH  Google Scholar 

  12. O. A. Shcherbina, “Asymptotic estimates of the effectiveness of local algorithms in discrete programming,” Zh. Vychisl. Mat. Mat. Fiz. 22, 1360–1368 (1982).

    MathSciNet  MATH  Google Scholar 

  13. D. V. Lemtyuzhnikova and D. V. Kovkov, “Testing of algorithms for integer quasiblock optimization problems,” Vestn. MGTU Baumana, Ser. Inform. Tekhnol., No. 1, 59–75 (2018).

  14. V. V. Voloshinov, D. V. Lemtyuzhnikova, and V. I. Tsurkov, “Grid parallelizing of discrete optimization problems with quasi-block structure matrices,” J. Comput. Syst. Sci. Int. 56, 930 (2017).

    Article  MathSciNet  MATH  Google Scholar 

  15. A. A. Mironov, V. V. Fedorchuk, and V. I. Tsurkov, “Minimax in transportation models with integral constraints: II,” J. Comput. Syst. Sci. Int. 44, 732 (2005).

    MATH  Google Scholar 

  16. A. A. Mironov, V. V. Fedorchuk, and V. I. Tsurkov, “Minimax in transportation models with integral constraints: I,” J. Comput. Syst. Sci. Int. 42, 562 (2003).

    MATH  Google Scholar 

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Funding

This work was supported by the Russian Foundation for Basic Research, project no. 18-31-00458.

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Authors and Affiliations

  1. Trapeznikov Institute of Control Sciences, Russian Academy of Sciences, 117997, Moscow, Russia

    D. V. Lemtyuzhnikova

  2. Moscow Aviation Institute (National Research University), 125993, Moscow, Russia

    V. Yu. Leonov

  3. Federal Research Center “Computer Science and Control”, Russian Academy of Sciences, 119333, Moscow, Russia

    V. Yu. Leonov

Authors
  1. D. V. Lemtyuzhnikova
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  2. V. Yu. Leonov
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Correspondence to D. V. Lemtyuzhnikova.

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Translated by A. Klimontovich

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Lemtyuzhnikova, D.V., Leonov, V.Y. Large-Scale Problems with Quasi-Block Matrices. J. Comput. Syst. Sci. Int. 58, 571–578 (2019). https://doi.org/10.1134/S1064230719040099

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  • DOI: https://doi.org/10.1134/S1064230719040099

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