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.
Similar content being viewed by others
REFERENCES
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.
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).
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).
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).
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.
A. I. Golikov and Yu. G. Evtushenko, “Search for normal solutions in linear programming problems,” Comput. Math. Math. Phys. 40 (12), 1694–1714 (2000).
O. L. Mangasarian, “A Newton method for linear programming,” J. Optim. Theory Appl. 121 (1), 1–18 (2004).
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).
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).
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).
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).
J. E. Bobrow, “A direct minimization approach for obtaining the distance between convex polyhedra,” Int. J. Rob. Res. 8 (3), 65–76 (1989).
O. L. Mangasarian, “A finite Newton method for classification,” Optim. Methods Software 17 (5), 913–929 (2002).
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).
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).
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).
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).
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).
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].
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).
O. Axelsson, “A class of iterative methods for finite element equations,” Comput. Methods Appl. Mech. Eng. 9, 123–137 (1976).
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).
L. Yu, J. P. Barbot, G. Zheng, and H. Sun, “Compressive sensing with chaotic sequence,” IEEE Signal Proc. Lett. 17 (8), 731–734 (2010).
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
Corresponding authors
Additional information
Translated by I. Ruzanova
Rights and permissions
About this article
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
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0965542519120091