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

  • Irina Sokolinskaya
  • Leonid B. SokolinskyEmail author
Conference paper
Part of the Communications in Computer and Information Science book series (CCIS, volume 753)


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.


NSLP algorithm Non-stationary linear programming problem Large-scale linear programming Fejer mapping 


  1. 1.
    Chung, W.: Applying large-scale linear programming in business analytics. In: Proceedings of the 2015 IEEE International Conference on Industrial Engineering and Engineering Management (IEEM), pp. 1860–1864. IEEE (2015)Google Scholar
  2. 2.
    Tipi, H.: Solving super-size problems with optimization. Presentation at the meeting of the 2010 INFORMS Conference on O.R. Practice. Orlando, Florida, April 2010. Accessed 7 May 2017
  3. 3.
    Gondzio, J., et al.: Solving large-scale optimization problems related to Bells Theorem. J. Comput. Appl. Math. 263, 392–404 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Sodhi, M.S.: LP modeling for asset-liability management: a survey of choices and simplifications. Oper. Res. 53(2), 181–196 (2005)CrossRefzbMATHGoogle Scholar
  5. 5.
    Dyshaev, M.M., Sokolinskaya, I.M.: Predstavlenie torgovykh signalov na osnove adaptivnoy skol’zyashchey sredney Kaufmana v vide sistemy lineynykh neravenstv [Representation of trading signals based Kaufman adaptive moving average as a system of linear inequalities]. Vestnik Yuzhno-Ural’skogo gosudarstvennogo universiteta. Seriya: Vychislitel’naya matematika i informatika [Bull. South Ural State Univ. Ser. Comput. Math. Softw. Eng.] 2(4), 103–108 (2013)Google Scholar
  6. 6.
    Ananchenko, I.V., Musaev, A.A.: Torgovye roboty i upravlenie v khaoticheskikh sredakh: obzor i kriticheskiy analiz [Trading robots and management in chaotic environments: an overview and critical analysis]. In: Trudy SPIIRAN [SPIIRAS Proceedings], vol. 3, no. 34, pp. 178–203 (2014)Google Scholar
  7. 7.
    Radenkov, S.P., Gavryushin, S.S., Sokolyanskiy, V.V.: Avtomatizirovannyye torgovyye sistemy i ikh installyatsiya v rynochnuyu sredu (chast’ 1) [Automated trading systems and their installation in the market environment (Part 1)]. Voprosy ekonomicheskikh nauk [Probl. Econ.] 6(76), 70–74 (2015)Google Scholar
  8. 8.
    Dantzig, G.: Linear Programming and Extensions. Princeton University Press, Princeton (1998). 656 pp\({\rm {.}}\) zbMATHGoogle Scholar
  9. 9.
    Klee, V., Minty, G.J.: How good is the simplex algorithm? In: Proceedings of the Third Symposium on Inequalities, University of California, Los Angeles, CA, pp. 159–175. Academic Press, New York-London, 1–9 September 1969. Dedicated to the Memory of Theodore S. MotzkinGoogle Scholar
  10. 10.
    Khachiyan, L.G.: Polynomial algorithms in linear programming. USSR Comput. Math. Math. Phys. 20(1), 53–72 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Shor, N.Z.: Cut-off method with space extension in convex programming problems. Cybern. Syst. Anal. 13(1), 94–96 (1977)Google Scholar
  12. 12.
    Karmarkar, N.: A new polynomial-time algorithm for linear programming. In: Proceedings of the Sixteenth Annual ACM Symposium on Theory of Computing, pp. 302–311. ACM (1984)Google Scholar
  13. 13.
    Sokolinskaya, I.M., Sokolinskii, L.B.: Parallel algorithm for solving linear programming problem under conditions of incomplete data. Autom. Remote Control 71(7), 1452–1460 (2010)CrossRefzbMATHGoogle Scholar
  14. 14.
    Rechkalov, T.V., Zymbler, M.L.: Accelerating medoids-based clustering with the Intel many integrated core architecture. In: Proceedings of the 9th International Conference on Application of Information and Communication Technologies, Rostov-on-Don, Russia, pp. 413–417. IEEE, 14–16 October 2015Google Scholar
  15. 15.
    Zymbler, M.: Best-match time series subsequence search on the intel many integrated core architecture. In: Morzy, T., Valduriez, P., Bellatreche, L. (eds.) ADBIS 2015. LNCS, vol. 9282, pp. 275–286. Springer, Cham (2015). doi: 10.1007/978-3-319-23135-8_19 CrossRefGoogle Scholar
  16. 16.
    Sokolinskaya, I.M., Sokolinsky, L.B.: Implementation of parallel pursuit algorithm for solving unstable linear programming problems. Bull. South Ural State Univ. Ser. Comput. Math. Softw. Eng. 5(2), 15–29 (2016). doi: 10.14529/cmse160202. (in Russian)Google Scholar
  17. 17.
    Sokolinskaya, I., Sokolinsky, L.: Solving unstable linear programming problems of high dimension on cluster computing systems. In: Proceedings of the 1st Russian Conference on Supercomputing - Supercomputing Days 2015, Moscow, Russian Federation. CEUR Workshop Proceedings, vol. 1482, pp. 420–427., 28–29 September 2015Google Scholar
  18. 18.
    Eremin, I.I.: Fejerovskie metody dlya zadach linejnoj i vypukloj optimizatsii [Fejer Methods for Problems of Convex and Linear Optimization]. Publishing of the South Ural State University, Chelyabinsk (2009). 200 pp\(\rm {.}\) Google Scholar
  19. 19.
    Sokolinskaya, I., Sokolinsky, L.: Revised pursuit algorithm for solving non-stationary linear programming problems on modern computing clusters with manycore accelerators. In: Voevodin, V., Sobolev, S. (eds.) RuSCDays 2016. CCIS, vol. 687, pp. 212–223. Springer, Cham (2016). doi: 10.1007/978-3-319-55669-7_17 CrossRefGoogle Scholar
  20. 20.
    Eremin, I.I.: Teoriya lineynoy optimizatsii [The theory of linear optimization]. Publishing House of the “Yekaterinburg”, Ekaterinburg (1999). 312 pp\({\rm {.}}\) Google Scholar
  21. 21.
    Thiagarajan, S.U., Congdon, C., Naik, S., Nguyen, L.Q.: Intel Xeon Phi coprocessor developers quick start guide. White Paper. Intel (2013). Accessed 7 May 2017

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.South Ural State UniversityChelyabinskRussia

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