Advertisement

Parallel Method of Pseudoprojection for Linear Inequalities

  • Irina Sokolinskaya
Conference paper
Part of the Communications in Computer and Information Science book series (CCIS, volume 910)

Abstract

This article presents a new iterative method for finding an approximate solution of a linear inequality system. This method uses the notion of pseudoprojection which is a generalization of the operation of projecting a point onto a closed convex set in Euclidean space. Pseudoprojecting is an iterative process based on Fejer approximations. The proposed pseudoprojection method is amenable to parallel implementation exploiting the subvector method, which is also presented in this article. We prove both the subvector method correctness and the convergence of the pseudoprojection method.

Keywords

Linear inequality system Iterative method Fejer approximations Pseudoprojection Parallel algorithm Convergence 

References

  1. 1.
    Agmon, S.: The relaxation method for linear inequalities. Can. J. Math. 6, 382–392 (1954).  https://doi.org/10.4153/CJM-1954-037-2MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Motzkin, T.S., Schoenberg, I.J.: The relaxation method for linear inequalities. Can. J. Math. 6, 393–404 (1954).  https://doi.org/10.4153/CJM-1954-038-xMathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Merzlyakov, Y.I.: On a relaxation method of solving systems of linear inequalities. USSR Comput. Math. Math. Phys. 2, 504–510 (1963).  https://doi.org/10.1016/0041-5553(63)90463-4CrossRefzbMATHGoogle Scholar
  4. 4.
    Bregman, L.M.: The relaxation method of finding the common point of convex sets and its application to the solution of problems in convex programming. USSR Comput. Math. Math. Phys. 7, 200–217 (1967).  https://doi.org/10.1016/0041-5553(67)90040-7MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Gubin, L.G., Polyak, B.T., Raik, E.V.: The method of projections for finding the common point of convex sets. USSR Comput. Math. Math. Phys. 7, 1–24 (1967).  https://doi.org/10.1016/0041-5553(67)90113-9CrossRefGoogle Scholar
  6. 6.
    Germanov, M.A., Spiridonov, V.S.: On a method of solving systems of non-linear inequalities. USSR Comput. Math. Math. Phys. 6, 194–196 (1966).  https://doi.org/10.1016/0041-5553(66)90066-8CrossRefzbMATHGoogle Scholar
  7. 7.
    Goffin, J.L.: The relaxation method for solving systems of linear inequalities. Math. Oper. Res. 5, 388–414 (1980).  https://doi.org/10.1287/moor.5.3.388MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    González-Gutiérrez, E., Todorov, M.I.: A relaxation method for solving systems with infinitely many linear inequalities. Optim. Lett. 6, 291–298 (2012).  https://doi.org/10.1007/s11590-010-0244-4MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    González-Gutiérrez, E., Hernández Rebollar, L., Todorov, M.I.: Relaxation methods for solving linear inequality systems: converging results. TOP 20, 426–436 (2012).  https://doi.org/10.1007/s11750-011-0234-4MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Mandel, J.: Convergence of the cyclical relaxation method for linear inequalities. Math. Program. 30, 218–228 (1984).  https://doi.org/10.1007/BF02591886MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Konnov, I.: Combined Relaxation Methods for Variational Inequalities. LNE, vol. 495. Springer, Heidelberg (2001).  https://doi.org/10.1007/978-3-642-56886-2CrossRefzbMATHGoogle Scholar
  12. 12.
    Konnov, I.V.: A modified combined relaxation method for non-linear convex variational inequalities. Optimization 64, 753–760 (2015).  https://doi.org/10.1080/02331934.2013.820298MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Fejér, L.: Über die Lage der Nullstellen von Polynomen, die aus Minimumforderungen gewisser Art entspringen. In: Hilbert, D. (ed.) Festschrift, pp. 41–48. Springer, Heidelberg (1982).  https://doi.org/10.1007/978-3-642-61810-9_6CrossRefGoogle Scholar
  14. 14.
    Eremin, I.I.: Methods of Fejer’s approximations in convex programming. Math. Notes Acad. Sci. USSR 3, 139–149 (1968).  https://doi.org/10.1007/BF01094336MathSciNetCrossRefGoogle Scholar
  15. 15.
    Vasin, V.V., Eremin, I.I.: Operators and Iterative Processes of Fejér Type. Theory and Applications. Walter de Gruyter, Berlin, New York (2009)CrossRefGoogle Scholar
  16. 16.
    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).  https://doi.org/10.1007/978-3-319-55669-7_17CrossRefGoogle Scholar
  17. 17.
    Sokolinskaya, I.M.: Scalable algorithm for non-stationary linear programming problems solving. In: 2017 2nd International Ural Conference on Measurements (UralCon), pp. 49–53 (2017).  https://doi.org/10.1109/URALCON.2017.8120685
  18. 18.
    Sokolinskaya, I., Sokolinsky, L.B.: Scalability evaluation of NSLP algorithm for solving non-stationary linear programming problems on cluster computing systems. In: Voevodin, V., Sobolev, S. (eds.) RuSCDays 2017. CCIS, vol. 793, pp. 40–53. Springer, Cham (2017).  https://doi.org/10.1007/978-3-319-71255-0_4CrossRefGoogle Scholar
  19. 19.
    Sokolinskaya, I., Sokolinsky, L.B.: On the solution of linear programming problems in the age of big data. In: Sokolinsky, L., Zymbler, M. (eds.) PCT 2017. CCIS, vol. 753, pp. 86–100. Springer, Cham (2017).  https://doi.org/10.1007/978-3-319-67035-5_7CrossRefGoogle Scholar
  20. 20.
    Censor, Y., Elfving, T.: New methods for linear inequalities. Linear Algebra Appl. 42, 199–211 (1982).  https://doi.org/10.1016/0024-3795(82)90149-5MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    De Pierro, A.R., Iusem, A.N.: A simultaneous projections method for linear inequalities. Linear Algebra Appl. 64, 243–253 (1985).  https://doi.org/10.1016/0024-3795(85)90280-0MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Yang, K., Murty, K.G.: New iterative methods for linear inequalities. J. Optim. Theory Appl. 72, 163–185 (1992).  https://doi.org/10.1007/BF00939954MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Eremin, I.I.: Teoriya lineynoy optimizatsii [The theory of linear optimization]. Publishing House “Yekaterinburg”, Ekaterinburg (1999). (in Russian)Google Scholar
  24. 24.
    Sokolinsky, L.B.: Analytical estimation of the scalability of iterative numerical algorithms on distributed memory multiprocessors. Lobachevskii J. Math. 39, 571–575 (2018).  https://doi.org/10.1134/S1995080218040121MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.South Ural State UniversityChelyabinskRussia

Personalised recommendations