Three derivative-free projection methods for nonlinear equations with convex constraints

详细信息    查看全文

• 作者：Min Sun (1)
  Jing Liu (2)
  
  1. School of Mathematics and Statistics ; Zaozhuang University ; Shandong ; 277160 ; China
  2. School of Mathematics and Statistics ; Zhejiang University of Finance and Economics ; Hangzhou ; 310018 ; China
  
• 关键词：Nonlinear equations ; Derivative ; free method ; Projection method ; Global convergence
• 刊名：Journal of Applied Mathematics and Computing
• 出版年：2015
• 出版时间：February 2015
• 年：2015
• 卷：47
• 期：1-2
• 页码：265-276
• 全文大小：190 KB
• 参考文献：1. Zeidler, E.: Nonlinear Functional Analysis and Its Applications. Springer, New York (1990) CrossRef
  2. Wood, A.J., Wollenberg, B.F.: Power Generation, Operation, and Control. Wiley, New York (1996)
  3. Dirkse, S.P., Ferris, M.C.: MCPLIB: A collection of nonlinear mixed complementarity problems. Optim. Meth. Soft. 5, 319鈥?45 (1995) CrossRef
  4. Qi, L.Q., Tong, X.J., Li, D.H.: An active-set projected trust region algorithm for box constrained nonsmooth equations. J. Optim. Theory Appl. 120, 601鈥?25 (2004) CrossRef
  5. Tong, X.J., Zhou, S.Z.: A smoothing projected Newton-type method for semismooth equations with bound constraints. J. Ind. Manag. Optim. 1, 235鈥?50 (2005) CrossRef
  6. Ortega, J.M., Rheinboldt, W.C.: Iterative Solution of Nonlinear Equations in Several Variables. Academic Press, New York (1970)
  7. Ulbrich, M.: Nonmonotone trust-region method for bound-constrained semismooth equations with applications to nonlinear mixed complementarity problems. SIAM J. Optim. 11, 889鈥?17 (2001) CrossRef
  8. Tong, X.J., Qi, L., Yang, Y.F.: The Lagrangian globalization method for nonsmooth constrained equations. Comput. Optim. Appl. 33, 89鈥?09 (2006) CrossRef
  9. La Cruz, W., Raydan, M.: Nonmonotone spectral methods for large-scale nonlinear systems. Optim. Methods Softw. 18, 583鈥?99 (2003) CrossRef
  10. La Cruz, W., Martnez, J.M., Raydan, M.: Spectral residual method without gradient information for solving large-scale nonlinear systems of equations. Math. Comput. 75, 1449鈥?466 (2006) CrossRef
  11. Zhang, L., Zhou, W.J.: Spectral gradient projection method for solving nonlinear monotone equations. J. Comput. Appl. Math. 196, 478鈥?84 (2006) CrossRef
  12. Yu, Z.S., Lin, J., Sun, J., Xiao, Y.H., Liu, L.Y., Li, Z.H.: Spectral gradient projection method for monotone nonlinear equations with convex constraints. Appl. Numer. Math. 59, 2416鈥?423 (2009) CrossRef
  13. Yu, G.H., Niu, S.Z., Ma, J.H.: Multivariate spectral gradient projection method for nonlinear monotone equations with convex constraints. J. Ind. Manag. Optim. 9(1), 117鈥?29 (2013) CrossRef
  14. Wang, C.W., Wang, Y.J., Xu, C.L.: A projection method for a system of nonlinear monotone equations with convex constraints. Math. Meth. Oper. Res. 66, 33鈥?6 (2007) CrossRef
  15. Ma, F.M., Wang, C.W.: Modified projection method for solving a system of monotone equations with convex constraints. J. Appl. Math. Comput. 34, 47鈥?6 (2010) CrossRef
  16. Zhang, H.Y., Wang, Y.N., Wang, Y.J.: A projection and constraction method for a system of nonlinear equations with convex constraints. J. Qufu Norm. Univ. 35(2), 11鈥?6 (2009)
  17. Zheng, L.: A new projection algorithm for solving a system of nonlinear equations with convex constraints. Bull. Korean Math. Soc 50(3), 823鈥?32 (2013) CrossRef
  18. Karamardian, S.: Complementarity problems over cones with monotone and pseudomonotone maps. J. Optim. Theory Appl. 18(4), 445鈥?54 (1976) CrossRef
  19. Zarantonello, E.H.: Projections on Convex Sets in Hilbert Space and Spectral Theory. Academic Press, New York (1971)
  20. Cheng, W.Y.: A two-term PRP-based descent method. Numer. Funct. Anal. Optim. 28(11鈥?2), 1217鈥?230 (2007) CrossRef
  21. Dolan, E.D., Mor茅, J.J.: Benchmarking optimization software with performance profiles. Math. Program. 91, 201鈥?13 (2002) CrossRef
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Computational Mathematics and Numerical Analysis
  Applied Mathematics and Computational Methods of Engineering
  Theory of Computation
  Mathematics of Computing
  
• 出版者：Springer Berlin / Heidelberg
• ISSN：1865-2085

文摘

In this paper, we propose three derivative-free projection methods for solving nonlinear equations with convex constraints, which can be regarded as the combinations of some recently developed conjugate gradient methods and the well-known projection method. Compared with the existing derivative-free projection methods, we use some new hyperplanes to obtain the new iterate, and without the requirement of the Lipschitz continuity of the equation, we prove three new methods are globally convergent with an Armijo-type line search. Preliminary numerical results are reported to show the efficiency of the proposed methods.