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


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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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.


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

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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).

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  • 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