Abstract
In this paper, we propose a globally convergent method for solving constrained nonlinear systems. The method combines an efficient Newton conditional gradient method with a derivative-free and nonmonotone line search strategy. The global convergence analysis of the proposed method is established under suitable conditions, and some preliminary numerical experiments are given to illustrate its performance.
Similar content being viewed by others
References
Argyros, I.K., Hilout, S.: Estimating upper bounds on the limit points of majorizing sequences for Newton’s method. Numer Algorithms 62(1), 115–132 (2013)
Barzilai, J., Borwein, J.M.: Two-point step size gradient methods. IMA J. Numer Anal. 8(1), 141–148 (1988)
Beck, A., Teboulle, M.: A conditional gradient method with linear rate of convergence for solving convex linear systems. Math Methods Oper. Res. 59(2), 235–247 (2004)
Bellavia, S., Macconi, M., Morini, B.: An affine scaling trust-region approach to bound-constrained nonlinear systems. Appl. Num. Math. 44(3), 257–280 (2003)
Bellavia, S., Morini, B.: Subspace trust-region methods for large bound-constrained nonlinear equations. SIAM J. Numer. Anal. 44(4), 1535–1555 (2006)
Birgin, E.G., Krejić, N., Martínez, J.M.: Globally convergent inexact quasi-Newton methods for solving nonlinear systems. Numer. Algorithms 32(2), 249–260 (2003)
Bogle, I.D.L., Perkins, J.D.: A new sparsity preserving quasi-Newton update for solving nonlinear equations. SIAM J. Sci. Statist. Comput. 11(4), 621–630 (1990)
Broyden, C.G.: The convergence of an algorithm for solving sparse nonlinear systems. Math. Comp. 25, 285–294 (1971)
Cruz, W.L., Raydan, M.: Nonmonotone spectral methods for large-scale nonlinear systems. Optim Methods Softw. 18(5), 583–599 (2003)
Echebest, N., Schuverdt, M.L., Vignau, R.P.: A derivative-free method for solving box-constrained underdetermined nonlinear systems of equations. Appl. Math. Comput. 219(6), 3198–3208 (2012)
Ferreira, O.P., Gonçalves, M.L.N.: Local convergence analysis of inexact Newton-like methods under majorant condition. Comput. Optim Appl. 48(1), 1–21 (2011)
Floudas, C.A., et al.: Handbook of test problems in local and global optimization. In: Nonconvex Optimization and its Applications, vol. 33. Kluwer Academic, Dordrecht (1999)
Freund, R.: A transpose-free quasi-minimal residual algorithm for non-hermitian linear systems. SIAM J. Sci. Comput. 14(2), 470–482 (1993)
Freund, R., Grigas, P.: New analysis and results for the Frank-Wolfe method. Math. Program. pp. 1–32 (2014)
Freund, R.W., Nachtigal, N.M.: QMR: a quasi-minimal residual method for non-hermitian linear systems. Numer. Math. 60(1), 315–339 (1991)
Gonçalves, M.L.N.: Inexact gauss-Newton like methods for injective-overdetermined systems of equations under a majorant condition. Numer. Algorithms 72(2), 377–392 (2016)
Gonçalves, M.L.N., Melo, J.G.: A Newton conditional gradient method for constrained nonlinear systems. J. Comput. Appl Math. 311, 473–483 (2017)
Gonçalves, M.L.N., Oliveira, F.R.: An inexact Newton-like conditional gradient method for constrained nonlinear systems. Appl. Num. Math. 132, 22–34 (2018)
Jaggi, M.: Revisiting Frank-Wolfe: projection-free sparse convex optimization. In: Proceedings of the 30th International Conference On Machine Learning (ICML-13), vol. 28, pp 427–435 (2013)
Kanzow, C.: An active set-type Newton method for constrained nonlinear systems. In: Ferris, M.C., Mangasarian, O.L., Pang, J.-S. (eds.) Complementarity: Applications, Algorithms and Extensions, vol. 50 of Appl. Optim., pp 179–200. Springer (2001)
Kelley, C.: Iterative methods for linear and nonlinear equations. Society for Industrial and Applied Mathematics (1995)
Kozakevich, D.N., Martinez, J.M., Santos, S.A.: Solving nonlinear systems of equations with simple constraints. Comput. Appl Math. 16, 215–235 (1997)
La Cruz, W.: A projected derivative-free algorithm for nonlinear equations with convex constraints. Optim. Methods Softw. 29(1), 24–41 (2014)
La Cruz, W., Martínez, J.M., Raydan, M.: Spectral residual method without gradient information for solving large-scale nonlinear systems of equations. Math Comp. 75(255), 1429–1448 (2006)
Li, D.H., Fukushima, M.A.: Derivative-free line search and global convergence of Broyden-like method for nonlinear equations. Optim. Methods Softw. 13(3), 181–201 (2000)
Lukšan, L., Vlček, J.: Sparse and partially separable test problems for unconstrained and equality constrained optimization. Technical Report N. 767, Institute of Computer Science, Academy of Sciences of the Czech Republic (1999)
Lukšan, L., Vlček, J.: Test problems for unconstrained optimization. Technical Report N. 897, Institute of Computer Science, Academy of Sciences of the Czech Republic (2003)
Macconi, M., Morini, B., Porcelli, M.: Trust-region quadratic methods for nonlinear systems of mixed equalities and inequalities. Appl. Num. Math. 59(5), 859–876 (2009)
Marini, L., Morini, B., Porcelli, M.: Quasi-Newton methods for constrained nonlinear systems: complexity analysis and applications. Comput. Optim. Appl. 71, 147–170 (2018)
Martinez, M.J.: Quasi-inexact-Newton methods with global convergence for solving constrained nonlinear systems. Nonlinear Anal. 30(1), 1–7 (1997)
Moré, J.J., Garbow, B.S., Hillstrom, K.E.: Testing unconstrained optimization software. ACM Trans. Math Softw. 7(1), 17–41 (1981)
Morini, B., Porcelli, M., Toint, P.L.: Approximate norm descent methods for constrained nonlinear systems. Math. Comput. 87(311), 1327–1351 (2018)
Schubert, L.K.: Modification of a quasi-Newton method for nonlinear equations with a sparse Jacobian. Math. Comp. 24, 27–30 (1970)
Tsoulos, I.G., Stavrakoudis, A.: On locating all roots of systems of nonlinear equations inside bounded domain using global optimization methods. Nonlinear Anal. Real World Appl. 11(4), 2465–2471 (2010)
Wang, P., Zhu, D.: An inexact derivative-free Levenberg–Marquardt method for linear inequality constrained nonlinear systems under local error bound conditions. Appl. Math. Comput. 282, 32–52 (2016)
Zhang, Y., Zhu, D.-T.: Inexact Newton method via Lanczos decomposed technique for solving box-constrained nonlinear systems. Appl. Math Mech. 31(12), 1593–1602 (2010)
Zhu, D.: An affine scaling trust-region algorithm with interior backtracking technique for solving bound-constrained nonlinear systems. J. Comput. App. Math. 184(2), 343–361 (2005)
Acknowledgments
The authors would like to thank the three anonymous referees and the associate editor for their insightful comments on earlier drafts of this paper.
Funding
The work of these authors was supported in part by CAPES, FAPEG/CNPq/PRONEM-201710267000532, and CNPq Grants 302666/2017-6 and and 408123/2018-4.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Gonçalves, M.L.N., Oliveira, F.R. On the global convergence of an inexact quasi-Newton conditional gradient method for constrained nonlinear systems. Numer Algor 84, 609–631 (2020). https://doi.org/10.1007/s11075-019-00772-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11075-019-00772-0