Skip to main content

Newton-Type Method for Solving Systems of Linear Equations and Inequalities

Abstract

A Newton-type method is proposed for numerical minimization of convex piecewise quadratic functions, and its convergence is analyzed. Previously, a similar method was successfully applied to optimization problems arising in mesh generation. It is shown that the method is applicable to computing the projection of a given point onto the set of nonnegative solutions of a system of linear equations and to determining the distance between two convex polyhedra. The performance of the method is tested on a set of problems from the NETLIB repository.

This is a preview of subscription content, access via your institution.

REFERENCES

  1. A. I. Golikov and I. E. Kaporin, “Inexact Newton method for minimization of convex piecewise quadratic functions, in Numerical Geometry, Grid Generation, and Scientific Computing: Proceedings of the 9th International Conference, NUMGRID 2018/Voronoi 150, Celebrating the 150th Anniversary of G.F. Voronoi, Moscow, Russia, December 2018, Ed. by V. A. Garanzha, L. Kamenski, and H. Si, Lecture Notes in Computational Science and Engineering (Springer Nature, Switzerland AG, 2019). Vol. 131. https://doi.org/10.1007/978-3-030-23436-2_10.

  2. V. A. Garanzha and I. E. Kaporin, “Regularization of the barrier variational method of mesh generation,” Comput. Math. Math. Phys. 39 (9), 1426–1440 (1999).

    MathSciNet  MATH  Google Scholar 

  3. V. Garanzha, I. Kaporin, and I. Konshin, “Truncated Newton type solver with application to grid untangling problem,” Numer. Linear Algebra Appl. 11 (5–6), 525–533 (2004).

    MathSciNet  Article  Google Scholar 

  4. I. E. Kaporin, “Using inner conjugate gradient iterations in solving large-scale sparse nonlinear optimization problems,” Comput. Math. Math. Phys. 43 (6), 766–771 (2003).

    MathSciNet  Google Scholar 

  5. V. Garanzha and L. Kudryavtseva, “Hypoelastic stabilization of variational algorithm for construction of moving deforming meshes,” in Optimization and Applications: OPTIMA 2018, Ed. by Y. Evtushenko, M. Jacimovic, M. Khachay, Y. Kochetov, V. Malkova, and M. Posypkin, Communications in Computer and Information Science (Springer, Cham, 2019), Vol. 974, pp. 497–511. https://doi.org/10.1007/978-3-030-10934-9_35.

  6. A. I. Golikov and Yu. G. Evtushenko, “Search for normal solutions in linear programming problems,” Comput. Math. Math. Phys. 40 (12), 1694–1714 (2000).

    MathSciNet  MATH  Google Scholar 

  7. O. L. Mangasarian, “A Newton method for linear programming,” J. Optim. Theory Appl. 121 (1), 1–18 (2004).

    MathSciNet  Article  Google Scholar 

  8. A. I. Golikov, Yu. G. Evtushenko, and N. Mollaverdi, “Application of Newton’s method for solving large linear programming problems,” Comput. Math. Math. Phys. 44 (9), 1484–1493 (2004).

    MathSciNet  MATH  Google Scholar 

  9. 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).

    MathSciNet  Article  Google Scholar 

  10. B. V. Ganin, A. I. Golikov, and Yu. G. Evtushenko, “Projective-dual method for solving systems of linear equations with nonnegative variables,” Comput. Math. Math. Phys. 58 (2), 159–169 (2018).

    MathSciNet  Article  Google Scholar 

  11. S. Ketabchi, H. Moosaei, M. Parandegan, and H. Navidi, “Computing minimum norm solution of linear systems of equations by the generalized newton method,” Numer. Algebra Control Optim. 7 (2), 113–119 (2017).

    MathSciNet  Article  Google Scholar 

  12. J. E. Bobrow, “A direct minimization approach for obtaining the distance between convex polyhedra,” Int. J. Rob. Res. 8 (3), 65–76 (1989).

    Article  Google Scholar 

  13. O. L. Mangasarian, “A finite Newton method for classification,” Optim. Methods Software 17 (5), 913–929 (2002).

    MathSciNet  Article  Google Scholar 

  14. J. B. Hiriart-Urruty, J. J. Strodiot, and V. H. Nguyen, “Generalized Hessian matrix and second-order optimality conditions for problems with data,” Appl. Math. Optim. 11 (1), 43–56 (1984).

    MathSciNet  Article  Google Scholar 

  15. 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).

    MathSciNet  Google Scholar 

  16. A. I. Golikov and Yu. G. Evtushenko, “Regularization and normal solutions of systems of linear equations and inequalities,” Proc. Steklov Inst. Math. 289, Suppl. 1, 102–110 (2015).

    MathSciNet  Article  Google Scholar 

  17. O. Axelsson and I. E. Kaporin, “Error norm estimation and stopping criteria in preconditioned conjugate gradient iterations,” Numer. Linear Algebra Appl. 8 (4), 265–286 (2001).

    MathSciNet  Article  Google Scholar 

  18. I. E. Kaporin and O. Axelsson, “On a class of nonlinear equation solvers based on the residual norm reduction over a sequence of affine subspaces,” SIAM J. Sci. Comput. 16 (1), 228–249 (1995).

    MathSciNet  Article  Google Scholar 

  19. I. E. Kaporin and O. Yu. Milyukova, “A massively parallel preconditioned conjugate gradient algorithm for the numerical solution of systems of linear algebraic equations,” in Proceedings of the Department of Applied Optimization of the Computing Center of the Russian Academy of Sciences, Ed. by V. G. Zhadan (Vychisl. Tsentr Ross. Akad. Nauk, Moscow, 2011), pp. 32–49 [in Russian].

  20. M. R. Hestenes and E. L. Stiefel, “Methods of conjugate gradients for solving linear systems,” J. Res. Nat. Bur. Standards 49 (1), 409–436 (1952).

    MathSciNet  Article  Google Scholar 

  21. O. Axelsson, “A class of iterative methods for finite element equations,” Comput. Methods Appl. Mech. Eng. 9, 123–137 (1976).

    MathSciNet  Article  Google Scholar 

  22. J. Dongarra and V. Eijkhout, “Finite-choice algorithm optimization in Conjugate Gradients,” Lapack Working Note 159, University of Tennessee Computer Science Report UT-CS-03-502 (2003).

  23. L. Yu, J. P. Barbot, G. Zheng, and H. Sun, “Compressive sensing with chaotic sequence,” IEEE Signal Proc. Lett. 17 (8), 731–734 (2010).

    Article  Google Scholar 

Download references

ACKNOWLEDGMENTS

We are grateful to V.A. Garanzha for numerous helpful remarks that allowed us to substantially improve the presentation of the material in this paper.

Funding

This work was supported in part by the Russian Foundation for Basic Research, project no. 17-07-00510.

Author information

Authors and Affiliations

Authors

Corresponding authors

Correspondence to A. I. Golikov, Yu. G. Evtushenko or I. E. Kaporin.

Additional information

Translated by I. Ruzanova

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Golikov, A.I., Evtushenko, Y.G. & Kaporin, I.E. Newton-Type Method for Solving Systems of Linear Equations and Inequalities. Comput. Math. and Math. Phys. 59, 2017–2032 (2019). https://doi.org/10.1134/S0965542519120091

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0965542519120091

Keywords:

  • systems of linear equations and inequalities
  • regularization
  • penalty function method
  • duality
  • projection of a point
  • piecewise quadratic function
  • Newton’s method
  • preconditioned conjugate gradient method