Abstract
In this paper, based on the hyperplane projection technique, we propose a three-term conjugate gradient method for solving nonlinear monotone equations with convex constraints. Due to the derivative-free feature and lower storage requirement, the proposed method can be applied to the solution of large-scale non-smooth nonlinear monotone equations. Under some mild assumptions, the global convergence is proved when the line search fulfils the backtracking line search condition. Moreover, we prove that the proposed method is R-linearly convergent. Numerical results show that our method is competitive and efficient for solving large-scale nonlinear monotone equations with convex constraints.
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Wood, A.J., Wollenberg, B.F.: Power Generations, Operations, and Control. Wiley, New York (1996)
Meintjes, K., Morgan, A.P.: A methodology for solving chemical equilibrium systems. Appl. Math. Comput. 22, 333–361 (1987)
Meintjes, K., Morgan, A.P.: Chemical equilibrium systems as numerical test problems. ACM Trans. Math. Softw. 16, 143–151 (1990)
El-Hawary, M.E.: Optimal Power Flower: Solution Techniques, Requirement, and Challenges. IEEE Service Center, Piscataway (1996)
Dirkse, S.P., Ferris, M.C.: MCPLIB: a collection of nonlinear mixed complementarity problems. Optim. Methods Softw. 5, 319–345 (1995)
Dennis, J.E., Moré, J.J.: A characterization of superlinear convergence and its application to quasi-Newton methods. Math. Comput. 28, 549–560 (1974)
Dennis, J.E., Moré, J.J.: Quasi-Newton method, motivation and theory. SIAM Rev. 19, 46–89 (1977)
Dennis, J.E., Schnabel, R.B.: Numerical methods for unconstrained optimization and nonlinear equations, Prentice-Hall, Englewood Cliffs, NJ(1983)
Qi, L., Sun, J.: A nonsmooth version of Newton’s method. Math. Program. 58, 353–367 (1993)
Yamashita, N., Fukushima, M.: On the rate of convergence of the Levenberg-Marquardt method. Comput. Supplementum 15, 239–249 (2001)
Solodov, M.V., Svaiter, B.F.: A new projection method for variational inequality problems. SIAM J. Control Optim. 37, 765–776 (1999)
Wang, C.W., Wang, Y.J., Xu, C.L.: A projection method for a system of nonlinear monotone equations with convex constraints. Math. Methods Oper. Res. 66, 33–46 (2007)
Cheng, W.Y.: A PRP type method for systems of monotone equations. Math. Comput. Modell. 50, 15–20 (2009)
Polak, E., Ribière, G.: Note sur la convergence de directions conjugées”. Revue Francaise d’informatique et de recherche Opérationnelle, série rouge 16, 35–43 (1969)
Polyak, B.T.: The conjugate gradient mehod in extreme problems. USSR Comput. Math. Math. Phys. 9, 94–112 (1969)
Li, D.H., Wang, X.L.: A modified Fletcher-Reeves-type derivative-free method for symmetric nonlinear equations. Numer. Algebra Control Optim. 1, 71–82 (2011)
Zhang, L., Zhou, W., Li, D.H.: Global convergence of a modified Fletcher–Reeves conjugate gradient method with Armijio-type line search. Numer. Math. 104, 561–572 (2006)
Hager, W.W., Zhang, H.: A new conjugate gradient method with guaranteed descent and an efficient line search. SIAM J. Optim. 16, 170–192 (2005)
Xiao, Y., Zhu, H.: A conjugate gradient method to solve convex constrained monotone equations with applications in compressive sensing. J. Math. Anal. Appl. 405, 310–319 (2013)
Wang, X.Y., Li, S.J., Kou, X.P.: A self-adaptive three-term conjugate gradient method for monotone nonlinear equations with convex constraints. Calcolo 53, 133–145 (2016)
Beale, E.M.L.: A derivation of conjugate gradients. In: Lootsma, F.A. (ed.) Numerical Methods for Nonlinear Optimization, pp. 39–43. Academic Press, London (1972)
Nazareth, L.: A conjugate direction algorithm without line search. J. Optim Theory Appl. 23, 373–387 (1997)
Zhang, L., Zhou, W., Li, D.H.: A descent modified Polak-Ribière–Polyak conjugate gradient method and its global convergence. IMA J. Numer. Anal. 26, 629–640 (2006)
Zhang, L., Zhou, W., Li, D.H.: Some descent three-term conjugate gradient methods and their global convergece. Optim. Methods Softw. 22, 697–711 (2007)
Cheng, W.: A two-term PRP-based descent method. Numer. Funct. Anal. Optim. 28, 1217–1230 (2007)
Zhang, L., Zhou, W.: Spectral gradient projection method for solving nonlinear monotone equations. Comput. Appl. Math. 196, 478–484 (2006)
La Cruz, W., Martinez, J.M., Raydon, M.: Spectral residual method without gradient information for solving large-scale nonlinear systems of equations. Math. Comput. 75, 1429–1448 (2006)
Zhou, W.J., Li, D.H.: Limited memory BFGS methods for nonlinear monotone equations. J. Comput. Math. 25, 89–96 (2007)
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The authors gratefully acknowledge the helpful comments and suggestions of the anonymous reviewers.
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This research was partially supported by the Chinese National Science Foundation (NSFC61561019 and NSFC11371384) and Chongqing Graduate Student Research Innovation Project (CYS14020).
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Gao, P., He, C. An efficient three-term conjugate gradient method for nonlinear monotone equations with convex constraints. Calcolo 55, 53 (2018). https://doi.org/10.1007/s10092-018-0291-2
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DOI: https://doi.org/10.1007/s10092-018-0291-2