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

Computational Mathematics and Mathematical Physics volume 59, pages 2017–2032 (2019)Cite this article

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.

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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.

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Authors and Affiliations

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

    A. I. Golikov, Yu. G. Evtushenko & I. E. Kaporin

  2. Moscow Institute of Physics and Technology (National Research University), 141700, Dolgoprudnyi, Moscow oblast, Russia

    A. I. Golikov, Yu. G. Evtushenko & I. E. Kaporin

Authors
  1. A. I. Golikov
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  2. Yu. G. Evtushenko
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  3. I. E. Kaporin
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Correspondence to A. I. Golikov, Yu. G. Evtushenko or I. E. Kaporin.

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Translated by I. Ruzanova

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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

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  • 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