Projective-Dual Method for Solving Systems of Linear Equations with Nonnegative Variables

  • B. V. Ganin
  • A. I. Golikov
  • Yu. G. Evtushenko


In order to solve an underdetermined system of linear equations with nonnegative variables, the projection of a given point onto its solutions set is sought. The dual of this problem—the problem of unconstrained maximization of a piecewise-quadratic function—is solved by Newton’s method. The problem of unconstrained optimization dual of the regularized problem of finding the projection onto the solution set of the system is considered. A connection of duality theory and Newton’s method with some known algorithms of projecting onto a standard simplex is shown. On the example of taking into account the specifics of the constraints of the transport linear programming problem, the possibility to increase the efficiency of calculating the generalized Hessian matrix is demonstrated. Some examples of numerical calculations using MATLAB are presented.


systems of linear equations with nonnegative variables regularization projection of a point duality generalized Newton’s method unconstrained optimization transport linear programming problem 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    A. I. Golikov and Yu. G. Evtushenko, “Solution method for large-scale linear programming problems,” Dokl. Math. 70 (1), 615–619 (2004).MATHGoogle Scholar
  2. 2.
    V. A. Garanzha, A. I. Golikov, Yu. G. Evtushenko, and M. Kh. Nguen, “Parallel implementation of Newton’s method for solving large-scale linear programs,” Comput. Math. Math. Phys. 49 (8), 1303–1317 (2009).MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    O. L. Mangasarian, “A Newton method for linear programming,” J. Optim. Theory Appl. 121, 1–18 (2004).MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    L. D. Popov, “Quadratic approximation of penalty functions for solving large-scale linear programs,” Comput. Math. Math. Phys. 47 (2), 200–214 (2007).MathSciNetCrossRefGoogle Scholar
  5. 5.
    L. D. Popov, “Experience in organizing hybrid parallel calculations in the Evtushenko–Golikov method for problems with block-angular structure,” Autom. Remote Control 75 (4), 622–631 (2014).MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    I. I. Eremin, “On quadratic and fully quadratic problems of convex programming,” Russ. Math. 42 (12), 20–26 (1998).MathSciNetMATHGoogle Scholar
  7. 7.
    M. Frank and P. Wolfe, “An algorithm for quadratic programming,” Naval Res. Logist. Quart. 3, 95–110 (1956).MathSciNetCrossRefGoogle Scholar
  8. 8.
    A. N. Tikhonov, “On ill-posed problems in linear algebra and a stable method for their solution,” Dokl. Akad. Nauk SSSR 163 (4), 591–594 (1965).MathSciNetGoogle Scholar
  9. 9.
    V. N. Malozemov and G. Sh. Tamasyan, “Two fast algorithms for projecting a point onto the canonical simplex,” Comput. Math. Math. Phys. 56 (5), 730–743 (2016).MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    G. Sh. Tamasyan, E. V. Prosolupov, and T. A. Angelov, “Comparative study of two fast algorithms for projecting a point to the standard simplex,” J. Appl. Ind. Math. 10 (2), 288–301 (2016).MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  • B. V. Ganin
    • 1
  • A. I. Golikov
    • 1
  • Yu. G. Evtushenko
    • 1
  1. 1.Dorodnitsyn Computing CenterFRC CSC RASMoscowRussia

Personalised recommendations