Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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A NEW PROJECTION ALGORITHM FOR SOLVING A SYSTEM OF NONLINEAR EQUATIONS WITH CONVEX CONSTRAINTS

  • Zheng, Lian (Department of Mathematics and Computer Science Yangtze Normal University)
  • Received : 2012.03.08
  • Published : 2013.05.31
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Abstract

We present a new algorithm for solving a system of nonlinear equations with convex constraints which combines proximal point and projection methodologies. Compared with the existing projection methods for solving the problem, we use a different system of linear equations to obtain the proximal point; and moreover, at the step of getting next iterate, our projection way and projection region are also different. Based on the Armijo-type line search procedure, a new hyperplane is introduced. Using the separate property of hyperplane, the new algorithm is proved to be globally convergent under much weaker assumptions than monotone or more generally pseudomonotone. We study the convergence rate of the iterative sequence under very mild error bound conditions.

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References

  1. S. P. Dirkse and M. C. Ferris, MCPLIB: A collection of nonlinear mixed complementarity problems, Optim. Methods Software 5 (1995), 319-345. https://doi.org/10.1080/10556789508805619
  2. M. E. EL-Hawary, Optimal Power Flow: Solution Techniques, Requirement and Challenges, IEEE Service Center, Piscataway, NJ, 1996.
  3. J. Y. Fan, Convergence rate of the trust region method for nonlinear equations under local error bound condition, Comput. Optim. Appl. 34 (2006), no. 2, 215-227. https://doi.org/10.1007/s10589-005-3078-8
  4. J. Y. Fan and J. Y. Pan, An improved trust region algorithm for nonlinear equations, Comput. Optim. Appl. 48 (2011), no. 1, 59-70. https://doi.org/10.1007/s10589-009-9236-7
  5. J. Y. Fan and Y. X. Yuan, On the quadratic convergence of the Levenberg-Marquardt method without nonsingularity assumption, Computing 74 (2005), no. 1, 23-39. https://doi.org/10.1007/s00607-004-0083-1
  6. D. R. Han, A hybrid entropic proximal decomposition method with self-adaptive strategy for solving variational inequality problems, Comput. Math. Appl. 55 (2008), no. 1, 101-115. https://doi.org/10.1016/j.camwa.2007.03.015
  7. Y. R. He, A new double projection algorithm for variational inequalities, J. Comput. Appl. Math. 185 (2006), no. 1, 166-173. https://doi.org/10.1016/j.cam.2005.01.031
  8. S. Karamardian, Complementarity problems over cones with monotone and pseudomonotone maps, J. Optim. Theory Appl. 18 (1976), no. 4, 445-454. https://doi.org/10.1007/BF00932654
  9. F. M. Ma and C.W.Wang, Modified projection method for solving a system of monotone equations with convex constraints, J. Appl. Math. Comput. 34 (2010), no. 1-2, 47-56. https://doi.org/10.1007/s12190-009-0305-y
  10. K. Meintjes and A. P. Morgan, A methodology for solving chemical equilibrium system, Appl. Math. Comput. 22 (1987), no. 4, 333-361. https://doi.org/10.1016/0096-3003(87)90076-2
  11. B. T. Polyak, Introduction to Optimization, Optimization Software Inc., Publications Division, New York, 1987(Translated from Russian, with a foreword by Dimitri P. Bertsekas).
  12. M. V. Solodov and B. F. Svaiter, A globally convergent inexact Newton method for systems of monotone equations, In: Fukushima M., Qi L. (eds) Reformulation: piecewise smooth, semismooth and smoothing methods. Kluwer, Holanda (1998), 355-369.
  13. X. J. Tong and L. Qi, On the convergence of a trust-region method for solving constrained nonlinear equations with degenerate solution, J. Optim. Theory Appl. 123 (2004), no. 1, 187-211. https://doi.org/10.1023/B:JOTA.0000043997.42194.dc
  14. C. W. Wang and Y. J. Wang, A superlinearly convergent projection method for constrained systems of nonlinear equations, J. Global Optim. 44 (2009), no. 2, 283-296. https://doi.org/10.1007/s10898-008-9324-8
  15. C. W. Wang, Y. J. Wang, and C. L. Xu, A projection method for a system of nonlinear monotone equations with convex constraints, Math. Methods Oper. Res. 66 (2007), no. 1, 33-46. https://doi.org/10.1007/s00186-006-0140-y
  16. A. J. Wood and B. F. Wollenberg, Power Generations, Operations, and Control, Wiley, New York, 1996.
  17. N. Xiu and J. Zhang, Some recent advances in projection-type methods for variational inequalities, J. Comput. Appl. Math. 152 (2003), no. 1-2, 559-585. https://doi.org/10.1016/S0377-0427(02)00730-6
  18. E. H. Zarantonello, Projections on Convex Sets in Hilbert Spaces and Spectral Theory, Academic Press, New York, 1971.
  19. J. L. Zhang and Y. Wang, A new trust region method for nonlinear equations, Math. Methods Oper. Res. 58 (2003), no. 2, 283-298. https://doi.org/10.1007/s001860300302

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