Abstract and Applied Analysis

A Globally Convergent Matrix-Free Method for Constrained Equations and Its Linear Convergence Rate

Min Sun and Jing Liu

Full-text: Open access

Abstract

A matrix-free method for constrained equations is proposed, which is a combination of the well-known PRP (Polak-Ribière-Polyak) conjugate gradient method and the famous hyperplane projection method. The new method is not only derivative-free, but also completely matrix-free, and consequently, it can be applied to solve large-scale constrained equations. We obtain global convergence of the new method without any differentiability requirement on the constrained equations. Compared with the existing gradient methods for solving such problem, the new method possesses linear convergence rate under standard conditions, and a relax factor γ is attached in the update step to accelerate convergence. Preliminary numerical results show that it is promising in practice.

Article information

Source
Abstr. Appl. Anal., Volume 2014 (2014), Article ID 386030, 6 pages.

Dates
First available in Project Euclid: 2 October 2014

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1412277002

Digital Object Identifier
doi:10.1155/2014/386030

Mathematical Reviews number (MathSciNet)
MR3214429

Zentralblatt MATH identifier
07022283

Citation

Sun, Min; Liu, Jing. A Globally Convergent Matrix-Free Method for Constrained Equations and Its Linear Convergence Rate. Abstr. Appl. Anal. 2014 (2014), Article ID 386030, 6 pages. doi:10.1155/2014/386030. https://projecteuclid.org/euclid.aaa/1412277002


Export citation

References

  • E. Zeidler, Nonlinear Functional Analysis and Its Applications, Springer, 1990.
  • A. J. Wood and B. F. Wollenberg, Power Generation, Operation, and Control, John Wiley and Sons, New York, NY, USA, 1996.
  • K. Meintjes and A. P. Morgan, “A methodology for solving chemical equilibrium systems,” Applied Mathematics and Computation, vol. 22, no. 4, pp. 333–361, 1987.
  • L. Qi, X. J. Tong, and D. H. Li, “Active-set projected trust-region algorithm for box-constrained nonsmooth equations,” Journal of Optimization Theory and Applications, vol. 120, no. 3, pp. 601–625, 2004.
  • M. Ulbrich, “Nonmonotone trust-region methods for bound-constrained semismooth equations with applications to nonlinear mixed complementarity problems,” SIAM Journal on Optimization, vol. 11, no. 4, pp. 889–917, 2001.
  • J. M. Ortega and W. C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York, NY, USA, 1970.
  • C. Wang, Y. Wang, and C. Xu, “A projection method for a system of nonlinear monotone equations with convex constraints,” Mathematical Methods of Operations Research, vol. 66, no. 1, pp. 33–46, 2007.
  • F. Ma and C. Wang, “Modified projection method for solving a system of monotone equations with convex constraints,” Journal of Applied Mathematics and Computing, vol. 34, no. 1-2, pp. 47–56, 2010.
  • Z. Yu, J. Lin, J. Sun, Y. Xiao, L. Liu, and Z. Li, “Spectral gradient projection method for monotone nonlinear equations with convex constraints,” Applied Numerical Mathematics, vol. 59, no. 10, pp. 2416–2423, 2009.
  • L. Zheng, “A new projection algorithm for solving a system of nonlinear equations with convex constraints,” Bulletin of the Korean Mathematical Society, vol. 50, no. 3, pp. 823–832, 2013.
  • W. La Cruz and M. Raydan, “Nonmonotone spectral methods for large-scale nonlinear systems,” Optimization Methods & Software, vol. 18, no. 5, pp. 583–599, 2003.
  • L. Zhang and W. Zhou, “Spectral gradient projection method for solving nonlinear monotone equations,” Journal of Computational and Applied Mathematics, vol. 196, no. 2, pp. 478–484, 2006.
  • W. Cheng, “A PRP type method for systems of monotone equations,” Mathematical and Computer Modelling, vol. 50, no. 1-2, pp. 15–20, 2009.
  • L. Zhang, W. Zhou, and D.-H. Li, “A descent modified Polak-Ribière-Polyak conjugate gradient method and its global convergence,” IMA Journal of Numerical Analysis, vol. 26, no. 4, pp. 629–640, 2006.
  • W. La Cruz, J. M. Martínez, and M. Raydan, “Spectral residual method without gradient information for solving large-scale nonlinear systems of equations,” Mathematics of Computation, vol. 75, no. 255, pp. 1429–1448, 2006. \endinput